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Definition of the Subject
The concept of Quantum Similarity (QS) was introduced for the first time in 1980 in a paper by Carbó et al. [1] entitled: How Similar is a Molecule to Another? There the basic aspects ofthe theory were set up. The backbone of QS was constituted by the conceptual support of the QSM.
QSM.
A QSM between two quantum objects (QO): associated to the density function (DF) tags: \( { \{\rho_A ,\rho_B\} } \) was defined as the density overlap integral:
which is always a positive real number, because the involved DF are non-negative definite real functions. Self-similarity measure integrals were defined in the same manner:
QS Matrix.
A simple example, which remains formally valid for larger QO sets, can illustrate the basic procedures of QS. For a set of two QO a symmetric QS matrix can be set up:
The columns (or rows) of the QS matrix:
can be interpreted here as the construction of a two dimensional discrete representation of the DF tags pair, quantum mechanically associated to the two involved QO. In this way one can write the association:
QSI.
From the QS matrix elements several kinds of QS indices (QSI) can also be described. Two indices were described in the seminal paper:
-
A)
Carbó Index; defined as a cosine of the angle subtended by the pair of DF tags \( { \{ \rho_A ,\rho_B\} } \):
$$ R_{AB} = Z_{AB} (Z_{AA} Z_{BB})^{-\frac{1}{2}} \in [0;1] \to R_{AB} = R_{BA} \:. $$ -
B)
Euclidean Distance Index; constructed with the well-known form:
$$ \begin{aligned} D_{AB}^2 &=\int_D | \rho_A (\mathbf{r}) - \rho_B (\mathbf{r})|^2 \mskip2mu\mathrm{d}\mathbf{r}\\ &= Z_{AA} + Z_{BB} -2 Z_{AB} \in [0;+\infty]\\ &\to D_{AB} =D_{BA}\:.\end{aligned} $$
Ordering a QOS and Mendeleyev Conjecture.
QSM or QSI, computed as previously described on the elements of a Quantum Object Set (QOS), are sufficient to permit ordering the elements of the set; thus, opening the way to construct non-artisan periodic tables of molecular sets, for instance.
The possibility to order QOS by means of their DF tags opened the way to estimate unknown properties of some QO; provided a set of QO with known properties was previously ordered. This has led to the path towards Mendeleyev conjecture , described by Carbó and Besalú [2] in 1996.
QQSPR.
Mendeleyev conjecture opened the way to employ QS techniques in order to obtain QSPR (Quantitative Structure-Properties Relationships), providing the necessary background for the description of quantum QSPR (QQSPR); a new kind of QSPR functionals that possess the properties to be: (1) Universal: as it can be employed to study any QOS. (2) Unbiased: as the user cannot choose any other QO descriptor than the DF tag. (3) Causal: as the QQSPR functionals are based on a non-empirical equation, derived from the application of quantum theoretical methodology.
Aims.
This contribution pretends to provide a mathematical basis for the understanding of the quantum QSPR problem, which tries to find out how to construct universal, unbiased and causal QSPR models. In turn these QSPR models can be employed to predict complex (in the sense of complicated observables) molecular properties. The ultimate purpose of such a theoretical framework is aimed at overcoming the fact that in previous applications molecular quantum similarity numerical values, ordered in the form of similarity matrices, have been employed just like molecular descriptors within a classical QSPR computational way. The future of molecular quantum similarity must be foreseen within the description and further development of an autonomous QQSPR set of computational procedures.
Introduction
Quantum Object Sets and Core Sets.
QSPR studies are based on some set of molecules: M, attached to a collection of descriptors and properties; the whole is the core set, symbolized as: C. For all the elements of the set M, via the Schrödinger equation for every molecule in the core set a wave function can be computed, providing in turn a set of density functions: \( { \text{P}=\rho_I } \), which can be considered as unique continuous molecular descriptors, by means of the quantum mechanical interpretation [3,4].
According to this it can be considered that:
Therefore, quantum similarity theory [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22], permits us to perform a Cartesian product of the M and P sets, which is used to build up a tagged set [23] \( { \text{Q}=\text{M}\times\text{P} } \), named a quantum object set (QOS). In a QOS the molecules constitute the object set and the density functions the tag set [23,24,25]. The elements of a QOS are quantum objects (QO). QO ordered pairs, are constructed in the following way:
Then the core set, C is a well-defined QOS. Because of the QOS definition in Eq. (1), C can have the form:
? contains the properties of some elements of M. Hence, C elements are triples made of molecular structures, density functions and properties: \( { {\text{C}}={\text{M}}\times {\text{P}}\times \Pi } \).
In classical QSPR density functions are replaced by finite dimensional vectors, whose elements are the so-called molecular descriptors. Construction of the core set with discrete vector spaces, substituting P, will also appear within the QQSPR. The substitution of the continuous density tags by discrete vectors in QQSPR has a mathematical-theoretical meaning, while in empirical QSPR this remains arbitrary. The elements of the core set C are core molecules, C-molecules or briefly C-m.
QQSPR Operators, Quantum Similarity Measures and the Fundamental QQSPR Equation.
Correspondence principle provides the rules to construct Hermitian operators, with expectation values associated to the experimental outcomes of submicroscopic systems observables [3,4]. For some complex (complicated) observables, like biological activities, the correspondence principle cannot be applied, as Hermitian operators are unavailable or difficult to be obtained. The QQSPR operators and the attached fundamental QQSPR equation, create an approximate quantum mechanical computational environment in order to estimate the expectation values of complicated observables.
QQSPR Operators.
The fundamental QQSPR equation arises when density function tags: \( { \{\rho_I(\mathbf{r})\} } \) of some QOS are used to construct a QQSPR operator as:
in Eq. (2), x 0 is an arbitrary constant; \( \{\Theta_\omega(\mathbf{R})| \omega = 0,1,2,\ldots\} \) is a known positive definite operator set, acting as a weight set; and \( { \{x_I\} } \) is a set of parameters, determined through the fundamental QQSPR equation.
The structure of a QQSPR operator (2) has to be seen as a first step algorithm permitting us to define approximate quantum mechanical operators. The QQSPR operators can be employed afterwards to evaluate their quantum mechanical expectation values.
Expectation Values of the QQSPR Operator.
To determine the parameter set \( { \{x_I\} } \), it is necessary to compute the set of expectation values over the elements, the core molecules or C-m, of the core set C. Besides a well-defined structure and a known density function, as members of a QOS, the C-m possess a known property value of the set: \( { \Pi =\{x_I\} } \), attached to each one.
Then, every known property of the C-m elements can be expressed as an expectation value of a QQSPR operator:
Zero-th Order Term.
In the expectation values (3) of the elements of C, the Zero-th order term is:
being a constant for each C-m, the Zero-th order term: \( { x_0 \Theta_0 (\mathbf{r}) } \) acts as an origin shift. Choosing: \( { \Theta_0(\mathbf{r})=I } \), this term becomes proportional to the number of electrons of the C-m considered:
The Zero-th order term can be omitted if it is no longer necessary to shift the property values of the C-m.
Quantum Similarity Measures in First and Second-Order Expectation Value Terms.
The first-order term in Eq. (3) contains QSM integralsbetween pairs of C-m density function tags, long time known [15]:
and the second-order term is made of triple-density quantum similarity measures [16]
The matrix symbol Z will be used to represent any collection of QSM: \( { \{z_{IJ}\} } \), independently of the nature of the weighting operator T1.
Quantum Similarity Matrices (QSM) in the Construction of First-Order QSPR Operators and the Definition of Discrete QOS (DQOS).
The first-order approach of the QSPR operator [13,14,15,16,17], applied to the core set with the known property set: \( { \Pi = \{\pi_I\} } \), produces the equation collection:
If \( { \Theta_1 =I } \) is used, the first-order integrals (4) are:
and can be ordered into a \( { (n\times n) } \) symmetric array, constructing in this way the so-called quantum similarity matrix : \( { \mathbf{Z}=\{z_{IJ}\} } \) (QS Matrix) [18]. The property set form a column vector: \( { | \mathbf{p} \rangle = \{p_I\} } \). Equations (4) are a linear system, which can be used to evaluate unknown molecular properties for some QOS members of the core set.
Every column of the QSM [19]: \( { \mathbf{Z} =\{ | \mathbf{z}_I \rangle = \{z_{JI}\}\} } \), is a discrete representation of each QO density function in: \( { P=\{\rho_I\} } \). A one-to-one correspondence exists between the density tag set and the QSM column submatrices:
The QSM column set can be used as a n-dimensional vector tag set, attached to the molecular set, building up a tagged set, called discrete quantum object set (DQOS) [19,20,21,22,26]:
In DQOS, the density function tags of the original QOS, Q, belonging to the tag set P, are substituted by the columns of the QSM. There also exists a one-to-one correspondence between both QOS: \( { Q\leftrightarrow Q_Z } \).
Fundamental QQSPR Equation Setup.
Expectation values of the QQSPR operator (3) can be collected in a column vector, providing the fundamental QQSPR equation :
In (6), \( { |\mathbf{p}\rangle =\{p_K\} } \) is the shifted C-m properties vector: \( { |\mathbf{p}\rangle =|\pi\rangle -|\theta\rangle } \), where \( { |\pi\rangle = \{\pi_I\} } \) is the original property vector and \( { |\theta \rangle = \{\theta_K\} } \) is the completely determined Zero-th order origin shift vector, \( { \{\mathbf{Z}_\omega| \omega =1,2,\ldots\} } \) is a matrix set containing the quantum similarity measures, for instance: \( { Z_1 =\{z_{IK}\}; Z_2 =\{z_{IJK}\};\ldots } \), and: \( { |\mathbf{x} \rangle = \{x_I\} } \) is a column vector bearing the unknown coefficients, which define explicitly the QQSPR operator (2).
The unknown coefficients \( { |\mathbf{x}\rangle = \{x_I\} } \) can be obtained solving the linear equation contained in the fundamental QQSPR Eqs. (4):
Equation (7) has no predictive power. This is so because the first-order similarity matrix \( { \mathbf{Z}_1 } \) has to be chosen positive definite by construction. By predictive power it is understood the possible computation of the property value for an also known quantum object, a U-m, which as such possesses a well-defined structure and density function, but belongs to the U set.
In the last years, since the description of QS measures for the first time [1], the predictive power of the information contained in the QS matrices set has been manipulated within the classical QSPR. This is the same as considering the similarity matrices as a source of molecular parameters to construct empirical QSPR. See references [5] and [27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46].
First-Order Fundamental QQSPR Equation.
The study of QQSPR predictive potential starts with the first-order fundamental QQSPR equation, involving the core set, containing the associated DQOS molecules, possessing known values of some complex property.
The first-order QQSPR fundamental equation in a compact matrix form [60] is written as:
Where the matrix Z is the already described symmetric QSM, \( { |\mathbf{p}\rangle } \) is the known core set shifted property vector and \( { |\mathbf{x}\rangle } \) is a \( { (n\times 1) } \) vector, whose coefficients have to be evaluated.
The predictive power of such an equation is a priori null. This is so because the QSM: Z, is by construction non-singular, then exists a QSM inverse \( { \mathbf{Z}^{-1} } \), with the relationships: \( { \mathbf{Z}^{-1} \mathbf{Z} = \mathbf{ZZ}^{-1} = \mathbf{I} } \). This leads to the trivial result, defining the unknown coefficient vector:
And exact property values for any molecule of the core set, can be reproduced just choosing the scalar products:
QSM for varied core sets have been used in a set of prediction studies [27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46], employing up to date statistical tools, typical of classical QSPR studies, [47,48,49,50,51,52,53,54,55,56,57,58]. The use of the fundamental QQSPR equation to construct algorithms, which can be utilized as predictive tools independently of classical QSPR algorithms, has been previously attempted [59], but it has not been continued in practice until recently [60,61,62]. Here will be discussed in several places not only the QQSPR problem itself, but various points of view and a future perspective as well.
Symmetrical Similarity Matrices.
The fundamental QQSAR equation has been presented within the particular case where the basis and probe molecular QOS coincide, forming a square symmetric QS Matrix. This choice has the drawback that the fundamental QQSPR linear system becomes well defined, with a unique solution, whenever the similarity matrix is non-singular as no QO coincides with another within the QOS C. Even then, there is quite a wide range of solutions to overcome this apparent limitation. Among other procedures, one can use the C symmetric QSM, Z, as a source of molecular descriptors and afterwards employ them in classical statistical treatments. This choice, as has been already commented, has been studied in many publications of our laboratory with success [27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46]. Other possible QSM can be constructed bearing rectangular structure, for example using known molecular structures acting as a basis set, which in turn can be compared with the core set.
Origin of Hansch QSAR Models .
An interesting possibility of the symmetric square representation for the QS Matrices corresponds to its potential to unveil the origin of one parameter classical QSAR models, such as those Hansch [63] described some years ago. A fundamental QQSPR linear equation can be associated to a set of ? equations with the same number of unknowns, and can be rewritten as:
There is no need to attach the QSM elements to any specific QOS, as all of them are computed over a unique basis of density function tags. Considering (11), two terms can be seen. The first one is attached to a self-similarity measure \( { z_{JJ} } \), while the second term in cases of some not so strongly varying QOS, can be considered almost a constant, that is using:
Equation (11) takes the final form:
which has the required appearance to be considered as possessing a Hansch-like structure.
Equation (13) proves self-similarities can be substitutes of the classical Hansch analysis parameters [63]. They constitute, for co-generic QOS, molecular descriptors with the property to be directly attached to a tri-dimensional molecular structure. Quantum self-similarity measures vary slowly with conformational changes [10,64], so their values for the optimal molecular geometry can be safely used.
Mathematical Background of Quantum Similarity
Inward Matrix Product (IMP)
IMP Definition.
An essential piece of QSM theory is the matrix operation called the inward matrix product (IMP) [65,66,67,68,69], which is based on the structure of the Hadamard product [70]Footnote 1. Such an operation is an internal composition law, which can be defined within a matrix (or hypermatrix) vector space \( { M_{(m\times n)} (K) } \) of arbitrary dimension \( { (m\times n) } \) and defined over a field K, producing a matrix whose elements are products made in turn by the elements of the matrices appearing in the IMP itself, according to the straightforward algorithm:
IMP is an operation, which can be applied not only to matrix spaces but over a wide variety of mathematical objects, producing another mathematical object of the same kind as the ones involved in the operation.
IMP Properties.
On the other hand, IMP is equivalent to a feature involving arrays, present in high-level computer languages such as Fortran 95 [71], so practical programming of the IMP properties and characteristics is straightforward. IMP is commutative, associative, and distributive with respect to the matrix sum. Moreover, it has a multiplicative neutral element, the unity matrix, which has been customarily represented by a bold real unit symbol and formally defined as: \( { \mathbf{1} = \{1_{ij} = 1\} } \).
IMP Powers and Functions.
By an IMP power over a matrix \( { \mathbf{A} = \{a_{ij}\} } \), noted as: \( { \mathbf{A}^{[p]} } \) is understood the matrix whose elements are the corresponding powers of the elements of A, that is: \( { \mathbf{A}^{[p]} = \big\{a_{ij}^p\big\} } \). In the same manner, an IMP function of a matrix, noted as: \( { f[\mathbf{A}] } \), is defined as the matrix whose elements are the functions of the original matrix: \( { f[\mathbf{A}] = \{f(a_{ij})\} } \).
Scalar Product as an IMP Composite Operation.
A useful application example of IMP is associated to the total sum of the elements of an arbitrary matrix, \( { \mathbf{A} = \{a_{ij}\}\in M } \), by means of the symbol:
Connecting this definition with IMP, one can easily write:
Then, it is simple to construct the scalar product of two matrices of the same dimension, symbolized here as:\( { \langle\mathbf{A}|\mathbf{B}\rangle } \), by means of the IMP structure:
In this way, the definition of distances and cosines of the angle between two matrices can be also outlined. For instance, the cosine of the angle subtended by two matrices can be written, according to Eq. (16), as:
Vector Semispaces (VSS)
A vector semispace (VSS) [23,24,25,72,73] is a vector space, where the additive group has been substituted by an additive semigroup. An additive semigroup [74] is an additive group without reciprocal elements, which is the same as to consider negative elements not present in VSS. A matrix VSS will be made by matrices whose elements are positive definite or semi-definite. QS matrix structures belong to positive definite VSS. This is the same as to consider the matrix elements forming a VSS constructed by positive definite real numbers, extracted in turn from the \( { \mathbf{R}^+ } \) half-line. All the elements of a matrix VSS are non-singular matrices from the IMP point of view, while any matrix possessing a zero element will be non-existent in a VSS, if this strict sense is adopted. A functional VSS can be constructed by positive definite functions over a given domain and lacking of the null function in order to comply with the strict VSS characteristics.
Minkowski Norms in VSS.
Because of the positive definite structure of the components of a VSS, the easiest way to define a norm within such a mathematical configuration is Minkowski's. In a matrix VSS one can write:
Meanwhile, in any general functional VSS, an equivalent form can also be defined:
As final information one can see that a Minkowski norm in \( { M(\mathbf{R}^+) } \), and, thus, the complete sum of a matrix elements, can be considered as a linear operator, that is:
s-Shell Structure in VSS.
Minkowski-like norms classify the VSS elements in subsets, the s-shells , \( { S(\sigma) } \), whose elements are defined by the value of such a norm:
The unit shell \( { S(1) } \) is a VSS subset, which can generate all the other VSS shells. The existence of this property can be easily constructed as follows:
Convex Conditions.
The idea underlying convex conditions, associated to a numerical set, a vector, a matrix, or a function, has been described since the initial work on VSS and the related questions [23,24,25,72,73]. By the convex conditions symbol : \( { K(\mathbf{x}) } \) is meant that the conditions:
hold simultaneously for a given mathematical object x. Convex conditions become the same as considering that the object belongs to the unit shell of some VSS. Then, for such kinds of elements:
Conversely, the following property holds over any element of the unit shell: \( { \forall \mathbf{x}\in S(1)\to K(\mathbf{x}) } \).
Convex Linear Combinations Within a s-Shell.
Given an arbitrary s-shell: \( { S(\sigma)\subset V(\mathbf{R}^+) } \), of some VSS, then convex linear combinations of the elements of the \( { \sigma- } \)shell produce a new element of \( { S\left(\sigma \right) } \). That is, suppose that the convex conditions:
hold on a known set of scalars \( { \{\gamma_I\} } \). Then, the following property will be fulfilled for any arbitrary subset of elements belonging to the chosen \( { \sigma- } \)-shell:
owing to the fact that the summation symbol, associated here to a Minkowski norm, is a linear operator, thus:
Such a property is related to the possibility of constructing approximate atomic and molecular DF, by means of convex linear combinations, using a basis set of structurally simpler functions, which shall belong to the same VSS s-shell. One of the possible technical options has been described in a series of papers, where the choice of the simplified functions, in atomic electronic density fitting, was made by sets of 1s GTO. The approach was termed atomic shell approximation (ASA) [75,76,77,78,79,80,81,82,83] and has been successfully employed, among other possibilities, to make the integral computation and molecular superposition inherent in molecular quantum similarity measures (MQSM) easier.
Generating Vector Spaces
Generating Symbols.
Any VSS s-shell structure can be supposedly generated by a conventional vector space (VS). Such VS can be defined over the complex or real fields. It can be additionally provided by convenience with a positive definite metric structure. Indeed, suppose such a VS, defined for the sake of generality over the complex field: \( { V(\mathbf{C}) } \). Then, from a very general point of view, the following algorithm can be envisaged:
Where the IMP: \( { x=v^\ast \ast v } \) has been used to construct the VSS vector. Then the following sequence:
holds and has been employed to set the form of Eq. (22).
The quantum mechanical image of the density function construction appears as a particular case of the definition attached to Eq. (22). In addition, from a complementary point of view, a symbol to briefly summarize Eq. (22) could be described. One can say compactly that the vector generates a VSS vector x, using a generating symbol : \( { R(v\to x) } \), whenever the sequence of relationships in Eq. (22) holds [23,24,25,72,73].
Probability Density Distributions.
From the quantum mechanical point of view, when a wave function \( { \Psi(\mathbf{r}) } \) is known, then the attached DF \( { \rho(\mathbf{r}) } \) is generated in the following way: \( { R(\Psi \to \rho) } \) [25].
Another interesting point to be noted is that any probability distribution, discrete or continuous, belongs to some VSS unit shell. Probability distributions can be generated by the conveniently normalized elements of some normed or metric space, in order that the resultant VSS element belongs to the unit shell, \( { S(1) } \). Potentially, in this way they can belong to any other s-shell: \( { \forall\rho(\mathbf{r})\in S(1)\wedge \sigma \in \mathbf{R}^+\to p(\mathbf{r})=\sigma \rho(\mathbf{r})\in S(\sigma) } \).
Thus, in VSS one can consider that the unit shell resumes every other s-shell. In this manner, probability distributions of any kind can be transformed into any other element of the associated VSS. One can also say that any VSS s-shell, \( { S(\sigma) } \), can be considered like a homothetic construct of the unit shell, \( { S(1) } \) [72]. Because the elements of the unit shell comply with the adequate form of a probability distribution, then they also fulfil the convenient convex conditions. That is: \( { \forall x\in S(1)\to K(x) } \). In other words, any probability distribution can be considered as an element of the unit shell forming part of a VSS.
Scalar Products and Measures in VSS.
Because of these possible connections attached to probability vectors, scalar products of two distinct compatible probability distributions are always positive definite, as one has:
Therefore, the cosine of the angle subtended by two probability distributions has to be contained in the open interval: \( { \left({0,1} \right] } \), due to Eq. (23). This is so, because the cosine of the subtended angle between two elements can be computed as:
Furthermore, Eq. (23) shows that the scalar product between two, or more, elements of the unit shell of a given metric VSS, constitute in any case a measure. In this way, such a scalar product can be considered a generalized volume.
Quantum Similarity
QSM over the Unit Shell.
A similarity measure over the unit shell of any VSS can be defined through the description of the mathematical elements described so far. Quantum similarity measures (QSM) were described a long time ago [1] and have been constantly used since then [84,85,86,87,88,89,90,91,92,93]. In the most simple and at the same time general way, within the easier formalism available, a QSM can be defined knowing the appropriate DF of two quantum systems: \( { \{\rho_A;\rho_B\} } \), adapted to the unit shell of the corresponding VSS, and using as weight some positive definite operator: \( { \{\Omega\} } \), then the integral measure:
will correspond to a weighted scalar product, defined over the unit shell elements, made in turn by the compatible quantum DF. The QSM (24) can be associated to a property very comparable to the one encountered in Eq. (24) and in any instance has to possess a positive definite nature.
Quantum Object Sets.
Suppose a set of quantum systems: \( { S=\{s_I\} } \), in a well-defined set of states. Suppose that to every quantum system there is attached a known state DF, forming the set: \( { P=\{\rho _I\} } \), belonging to the unit shell of some functional VSS. A tagged set [23,24,25] can be constructed, using the Cartesian product: \( { T=S\times P } \), where each element, \( { \tau_I \in T } \), is constructed by the ordered pair composition rule: \( { \tau_I =(s_I;\rho_I) } \), forming in this way a quantum object . The tagged set T constitutes a quantum object set , that is: \( { T=\{\tau_I\} } \). The QSM earlier defined in Eq. (24), can be interpreted, in turn, as a tensor product of the tag part of the QOS.
Quantum Object Sets and Core Sets.
The usual problem in all QSPR studies is customarily based on the previous knowledge of some molecular set: M, of cardinality n, such that the structures and properties of the set elements are known beforehand. From now on, one can refer to this collection of molecules, molecular descriptors and properties as the core set and name it as: C. With the molecular structures of the elements of the set M known, one can solve the Schrödinger equation associated to, in principle, the ground state of every molecule in the set and compute an attached set of density functions: \( { {\text{P}} = \{\rho_I\} } \), which can be connected to sole continuous molecular descriptors, according to the quantum mechanical usual custom; such that:
In terms of the theoretical settings related with quantum similarity, the Cartesian product of the molecular and the density function sets is used to construct a tagged set: \( { {\text{Q}} = {\text{M}} \times {\text{P}} } \). Such a tagged set has been formerly named a quantum object set, where the molecules constitute the object set and the density functions act as the tag set. Thus, from now on, one can consider the core set as a well-defined QOS. One can name the elements of a QOS as quantum objects. Therefore, ordered pairs, constructed in the following way, define any QO:
However, the core set shall be structured in an even extended manner. Starting from the QOS definition, the core set C has also to be associated with the following characteristic:
Where the set ? contains all the properties of the elements of the molecular set M. Hence, core set elements are well-defined triples consisting of molecular structures, density functions, and properties: \( { {\text{C}} = {\text{M}} \times {\text{P}} \times \Pi } \).
In classical QSPR, based on the empirical description of the set M, the set of density functions is replaced by a set of vectors, belonging to a finite dimensional space, whose elements are the chosen molecular descriptors. The possible construction of the core set within discrete vector spaces, substituting the density tag set P is a characteristic, which will also appear within the QQSPR theoretical development, as will be explained below. It must be said that the QQSPR substitution of the continuous density tags by discrete vectors has a quite well-structured mathematical-theoretical meaning, while in empirical QSPR remains arbitrarily chosen.
Similarity Matrices.
Collecting all the QSM computed between the element pairs of a given QOS, a so-called Quantum Similarity Matrix (QSM) is obtained, and constructed according to the definition (24) by means of: \( { \mathbf{Z} = \{z_{ij}\} } \). Because of the QS matrix elements structure, the matrix itself can be considered as an element of some VSS of the appropriate dimension. The QS matrix Z is a symmetric matrix with positive definite elements, whose columns \( { \{\mathbf{z}_I\} } \) (or rows) are also elements of some N-dimensional VSS. As such, there exists a real symmetric matrix, X, such that, in general \( { R(\mathbf{X} \to \mathbf{Z}) } \), that is:
As a consequence, any QS matrix belongs to a precise s-shell of some VSS. That is:
Stochastic Similarity Matrices .
Even if the columns or rows of the QS matrix Z belong to different s-shells of some VSS, they can be easily brought to the unit shell, by using a set of simple homothetic transformations, involving a product by a diagonal matrix, with elements constructed by the Minkowski norms of the columns (or rows) of the QS matrix. That is, the diagonal matrix:
can transform the QS matrix Z into a column (or row) stochastic matrix, simply by multiplying on the right (or the left) of Z by the inverse of D, respectively [59]. For instance, the stochastic column matrix associated to the QS matrix is:
In this way, the columns \( { \{s_I\} } \) of the stochastic matrix,S, belong to the unit shell of the column vector VSS of the appropriate dimension. That is:
However, the column stochastic QS matrix (28) appears to be no longer symmetric as his originating QS matrix Z is.
Quantum Similarity Matrix Aufbau Procedure [94]
Suppose a known given Quantum Object Set formed by N molecules, with density tags described as: \( { \{\rho_I(\mathbf{r})\} }\). Up to now, the usualprocedure to construct the QSM has been to maximize each of theintegrals of type (26) with respect to the translations and rotationsof one of the implied QO in relation to the others. This can beexpressed formally, for instance, as:
However, as has been well known since the first paper on the subject [1] the set of quantum similarity measures \( { \{z_{IJ}\} } \) depend on the relative position in 3D space of the implied Quantum Objects (QO's). As the QO density function labels are positive definite functions, the integrals of type (29) can be considered as measures; thus, they are positive definite too. Up to now the usual procedure to construct the QSM has been to maximize each of the integrals of type (29) with respect to the translations and rotations of one of the implied QO in relation to the other. This can be expressed formally, for instance, as:
where the pairs: \( { \{\mathbf{t};\boldsymbol{\Omega}\} } \) are translations and rotations respectively, performed on the center of coordinates of the Jth QO. It is irrelevant which one of the QO pair is chosen in order to optimize the integral (29) by means of the algorithm (30), the same result shall be obtained choosing the Ith QO for undertaking translations and rotations.
Apparently, such a procedure, repeated for every non-redundant couple of QO's, shall provide a QSM Z with appropriate characteristics associated to a metric matrix. The most important one is that the attached QSM has the property to be positive definite; as the density tag set is linearly independent, if the QOS is made of different QO's, then Z has to be a metric matrix of a pre-Hilbert space [94]. However, in many cases the use of algorithm (30) does not provide a QSM whose whole spectrum is positive definite, but a small amount of the Z eigenvalues may appear to be negative. This non-definite behavior of the metric matrix Z can be attributed to the fact that following algorithm (30), when facing the Jth QO to the rest of the QOS elements, then for every distinct QO a different relative position of the Jth QO is found, while reaching the optimal value of the similarity measure (29) for every pair of QO's; that is: the relative position of the Jth QO with respect to the Ith QO, \( { \forall I\colon J\ne I } \), in order to optimize every element z IJ , becomes different, and therefore when optimizing Eq. (30) one will obtain a set of different optimal translations-rotations: \( { \{\mathbf{t}_I;\boldsymbol{\Omega}_I\} \forall I\ne J } \).
When computing any optimal quantum similarity measure by means of algorithm (30), one also must be aware that the final result, can be used to construct the symmetric \( { (2\times 2) } \) matrix:
and also has to provide at least a positive definite matrix (31), which is the same as to consider the following property has to be fulfilled:
The restriction (32) can also be written as:
and this will provide a form of the \( { (2\times 2) } \) positive definite restrictions to be easily related to the general analysis which follows. Therefore, the algorithm (30) has to be modified accordingly incorporating the inequality (32) as a restriction:
and one can expect that the general QSM Z, can approach in this way the required complete positive definiteness, although this cannot be completely assured. In fact, this \( { (2\times 2) } \) restriction constitutes an incomplete point of view, as nothing can be said about the positive definiteness of higher dimensional submatrices of the QSM Z. In this sense, the restricted algorithm (34) is more or less similar to the triangle distance relationship coherence, sought by an already published procedure [91].
The Quantum Similarity Matrix Aufbau Recursive Algorithm.
Although one can use the Gershgorin theorem to test the positive definiteness of any QSM, a complete QSM calculation algorithm, based on the generalization of property (33) for \( { (2\times 2) } \) matrices, in order to assure the QSM Z positive definiteness, shall be based on an Aufbau procedure; that is: starting from any pair of QO, algorithm (34) is put forward. The result will be a positive definite matrix, \( { \mathbf{Z}_\mathbf{0} } \) say, with a structure like the matrix (31) defined above. A simple recursive Aufbau algorithm can be described in order to obtain a final positive definite QSM.
Suppose that for some index \( { P<N } \), a \( { (P\times P) } \) positive definite QSM \( { \mathbf{Z}_\mathbf{0} } \) has been obtained, using the QO's sequence: \( { \{I_K;K=1,P\} } \). One can add a new QO to the Aufbau procedure, the Qth QO, say, in such a way that an augmented QSM, \( { \mathbf{Z}_{\mathbf{1}} } \), is obtained possessing the partitioned structure:
with the \( { (1\times P) } \) row vector defined as: \( \langle\mathbf{z}| = (z_{I_1 Q}; z_{I_2 Q};\ldots;z_{I_P Q}) \), and the column vector \( { |\mathbf{z}\rangle } \), being just the transpose of the former; finally, \( { \theta \equiv z_{QQ} } \) is the self-similarity of the added QO.
The sufficient relationship, which can be written here as the set of conditions:
assuring that the augmented matrix \( { \mathbf{Z}_\mathbf{1} } \) has a positive definite structure, can be alternatively rewritten via a recursive Cholesky decomposition algorithm, described in several places [17,96].
The necessary and sufficient condition for the positive definiteness of the augmented QSM \( { \mathbf{Z}_\mathbf{1} } \) can be stated as:
The Cholesky decomposition condition, which can be called here Quantum Similarity Aufbau Condition (QSAC), means that it cannot be reliable to use a pair of QO's every time that a new element of the QSM has to be computed, but that the added QO density function: \( { \rho_Q(\mathbf{r}|\mathbf{t};\boldsymbol{\Omega}) } \) has to be translated-rotated with the same values of the pair: \( { \{\mathbf{t};\boldsymbol{\Omega}\} } \), for every computed element of the vector \( { |\mathbf{z}\rangle } \), connecting recursively the QO Q with all the ones previously employed in constructing the QS submatrix \( { \mathbf{Z}_\mathbf{0} } \). When the QS submatrix \( { \mathbf{Z}_\mathbf{0} } \) has scalar \( { (1\times 1) } \) dimension as occurs in the submatrix (31) case, then the QSAC (36) becomes the relationship (33). Moreover, the QSAC condition is a stronger positive definiteness condition than the diagonal dominance, as QSAC becomes the necessary and sufficient condition for constructing a positive definite augmented matrix.
The maximal pair condition (30) can be substituted in the general \( { (P\times P) } \) case, for instance, by maximizing the sum of the whole vector \( { |\mathbf{z}\rangle } \), which due to the positive definiteness of its elements is coincident with the search of a maximal Minkowski norm:
This can be done admitting the same translation-rotation sequence performed on every term of the vector \( { |\mathbf{z}\rangle } \) in Eq. (37), whenever such transformation increases the Minkowski norm. However, while the maximal value of the sum leading to the Gershgorin radius is searched as in the condition (37) of the previous sentence, the QSAC relationship (36) has to be equally tested and if not fulfilled the pair \( { \{\mathbf{t};\boldsymbol{\Omega}\} } \) rejected. Such a procedure will assure the positive definiteness of the QSM Z at the final step of the recursion and will provide the same relative position in the calculation of the quantum similarity measures for every recursively added QO.
Geometrical Interpretation of the QSAC.
Leaving apart the linear algebra concept of diagonal dominance which similarity matrices usually do not fulfill, the alternative Cholesky decomposition condition property leading to the QSAC, assuring in this manner the positive definite structure of the final QSM form and written as in Eq. (36), has a clear geometrical meaning. The positive definite quadratic form: \( { \big\langle\mathbf{z}\big|\mathbf{Z}_\mathbf{0}^{-\mathbf{1}}\big|\mathbf{z}\big\rangle \in \mathbf{R}^+ } \), is nothing else than the Euclidean norm of the vector \( { |\mathbf{z}\rangle } \) in the reciprocal metric space defined by the density tags: \( \{\rho_{I_K}(\mathbf{r});K=1,P\} \). The QO's tags are employed to form the QSM \( { \mathbf{Z}_\mathbf{0} } \), which because of the QSAC construction has been structured positive definite and acts accordingly as a metric matrix of a P-dimensional pre-Hilbert space. Since in the quadratic form appearing in Eq. (36), the inverse of the metric \( { \mathbf{Z}_\mathbf{0} } \) appears, the implicit Euclidean norm equivalent to the aforementioned quadratic form is computed in the metric reciprocal space with the matrix \( { \mathbf{Z}_0^{-1} } \), acting as a positive definite metric matrix, because: \( { Sp[\mathbf{Z}_\mathbf{0}] \in \mathbf{R}^+\to Sp\big[\mathbf{Z}_0^{-1}\big]\in\mathbf{R}^+ } \). Accordingly, the QSAC forces this Euclidean norm in the reciprocal P-dimensional pre-Hilbert space to be less than the self-similarity of the recursively added Qth QO.
This permits us to associate the described Quantum Similarity Aufbau procedure as an algorithm maximizing the Minkowski norm of each recursive column \( { \left| \mathbf{z} \right\rangle } \) of the QSM, submitted to the QSAC restriction which means that its Euclidean norm, computed in the recursive reciprocal pre-Hilbert space, remains less than the recursive QSM diagonal self-similarity elements.
Finally, the following points must be taken into account:
-
1.
Because it is not necessary to start the recursive QSAC with any a priori chosen QO, the final QSM will certainly depend on the QO recursive order chosen. Thus, there are just \( { N! } \) possible choices, each one producing an equally positive definite QSM. However, the ordering imposed by the self-similarity measures can be chosen as a way to reach a systematic QSM Aufbau. That is, if one calls the QSM diagonal self-similarity measures set computed on the QOS elements: \( { D(\mathbf{Z})=\{z_{II}\} } \), then the obvious choices are defined by the maximal ordering:
$$ z_{11} = \max_I [D(\mathbf{Z})] \to z_{22} = \max_I [D(\mathbf{Z})-z_{11}]\ldots $$or by the minimal:
$$ z_{11} = \mathop{\min}\limits_I [D(\mathbf{Z})] \to z_{22} = \mathop{\min}\limits_I [D(\mathbf{Z})-z_{11}]\ldots $$This ensures that the QO's will be ordered in decreasing or increasing complexity, while providing a generic reproducible way of computing QSM under QSAC premises.
-
2.
When constructing the QSM according to the proposed Aufbau procedure, it is well known that the overlap quantum similarity measures, as defined in Eq. (29), can be substituted by a more general form involving a positive definite operator: \( { \Omega(\mathbf{r}_\mathbf{1};\mathbf{r}_\mathbf{2}) } \); so, in general, the similarity measures can be described as the integral:
$$ z_{IJ} (\Omega) = \iint_D \rho_I(\mathbf{r}_{\mathbf{1}}) \Omega(\mathbf{r}_\mathbf{1};\mathbf{r}_{\mathbf{2}})\rho_J(\mathbf{r}_\mathbf{2})\mskip2mu\mathrm{d}\mathbf{r}_\mathbf{1} \mskip2mu\mathrm{d}\mathbf{r}_{\mathbf{2}}\:; $$while the positive definite operator choice ensures that the QSM, when constructed according to the equivalent QSAC, like the one depicted previously for overlap quantum similarity measures in Eq. (36), is positive definite. One just shall make the substitution: \( { z_{IJ} \leftarrow z_{IJ}(\Omega) } \).
-
3.
The QSAC is also valid for quantum similarity measures involving the off-diagonal terms of the density matrix.
Linear Quantum QSPR Fundamental Equation
Expectation Values
In quantum mechanics, the expectation value of some QO observable, associated in turn to some Hermitian operator W, is measured in the usual statistical way [97,98,99], using the tag part ? A of the corresponding QO:
At the same time, in general, the operator W can be decomposed as follows:
O bears a positive definite and known form. On the other hand, the operator: Q, can be approximately expressed in terms of an appropriate linear (or multilinear) combination of a known density function set \( { \{\rho_I\} } \), provided with the adequate variable count, in order to match the one of ? A , that is:
The structure associated to an operator like: W, permits us to construct expectation values of entangled or complicated observables of submicroscopic systems. Such entangled observables can be considered connected to experimental outcomes, like biological activity, whose Hermitian operator cannot be completely well defined. So, this way to proceed appears quite appropriate for cases where the complexity of the observed phenomenon does not possess a straightforward association with any known or easily describable Hermitian operator. One shall stress the fact that the set up (39) is not appropriate for well-defined observables like kinetic energy, Coulomb energy, dipole and multipole moments,…
Quantum QSPR Fundamental Equation
Substituting the expression of the operator, described by Eq. (39) into expectation value expression (38), one arrives at the following result, related to the corresponding QO:
which, after supposing that several QO or the whole elements of a QOS are considered, this result can be brought into the matrix form of a linear equation:
Where the column vector \( { |\pi\rangle } \) contains the collection of expectation values of the considered QOS, \( { \mathbf{Z}=\{z_{IA}(\Omega)\} } \) is a quantum similarity matrix and, finally, the column vector w collects the coefficients by which the operator Q is approximately expressed by means of Eq. (39).
Interpretation and Characteristics of the Quantum QSPR Fundamental Equation
The interpretation of the linear system (40) can proceed as follows. The vector \( { |\pi \rangle } \) can supposedly contain known values of a well-defined, but arbitrarily complicated observable property of the chosen QOS. The quantum similarity matrix Z can be computed, once the elements of the QOS are supposedly known. The coefficient vector w has to be determined.
Put in such terms, Eq. (40) has the same well-known structure as the usual classical QSPR problems. However, this fact constitutes a very important and crucial result: Because, it permits us to interpret the columns (or rows) of the quantum similarity matrix, Z, as being the QSM finite-dimensional, discrete, descriptors of every QO used in the study. These considerations are sufficient to allow us to name Eq. (40) the quantum QSPR (QQSPR) fundamental equation .
Characteristics of the QQSPR Fundamental Equation.
Unlike the problems present in classical QSPR models, the QQSPR fundamental equation has several characteristics lacking in the former usual equations, these are:
-
1.
Universal applicability, because Eq. (40) can be used to model any kind of QOS: nuclei, atoms, molecules …
-
2.
Unbiased background descriptor structures, because the QSM elements, forming the quantum similarity matrix, Z, appearing in Eq. (40), are not arbitrarily chosen by the user, among those belonging to a given descriptor pool, but appear as a consequence of the theory.
-
3.
Causal character, as the QQSPR models obtained are the result of solving a well-defined equation, as shown through the set up of Eq. (40), and are deducible from the general theoretical structure of quantum mechanics.
Quantum Similarity Matrices (QSM) in the Construction of First-Order QSPR Operators and the Definition of Discrete QOS.
The first-order approach of the QSPR operator, for the core set known molecular property tag set: \( { \Pi = \{\pi_I\} } \) generates the following equation collection:
The set of integrals:
appearing in Eqs. (197) can be ordered into a \( { (n\times n) } \) symmetric array, constructing in this way the so-called quantum similarity matrix : \( { \mathbf{Z} = \{z_{IJ}\} } \) (QSM). In turn, the ordered set of shifted properties: \( { \{p_I\} } \) can form a \( { (n\times 1) } \) column vector: \( { |\mathbf{p}\rangle = \{p_I\} } \). Therefore, the equation set (197) is simply a linear system, which will be discussed next, in order to describe its possible use for evaluating U-m unknown molecular properties.
Discrete QOS.
Every column of the QSM: \( \mathbf{Z}=\{|\mathbf{z}_I\rangle = \{z_{JI}\}\} \), can be interpreted as a discrete matrix representation of each QO density matrix, present within the density function tag set: \( { \text{P} = \{\rho_I\} } \). In this way a one-to-one correspondence can be established between the density tag set and the QSM column submatrices, which can be written as:
In other words, the QSM column set can be used as a new n-dimensional vector tag set, attached to the molecular set M, in order to build up a new tagged set, namely a discrete quantum object set :
In this DQOS, the density function tags of the original QOS, Q, belonging to the tag set P, are substituted by the columns of the QSM. Therefore, there also exists a one-to-one correspondence between both QOS: \( { {\text{Q}}\leftrightarrow {\text{Q}}_{\text{Z}} } \).
The Nature of the QSM Descriptors.
In both the quantum similarity matrix Z, or its stochastic column transformation S, the involved columns forming both matrices possess a very special character, besides the fact that they belong to some VSS of the appropriate dimension.
Starting from the QOS, where each QO is defined by the ordered pair of submicroscopic systems and state density functions:
then, when dealing with the construction of the QSM or the stochastic transformation (28), which one can consider expressed through the decomposition of its columns as: \( { \mathbf{Z} = \{\mathbf{z}_I\} } \) or \( { \textbf{S} = \{\mathbf{s}_I\} } \), it can be deduced that both matrices induce a new possible QOS, made with discrete N-dimensional tags, instead of the infinite dimensional density function ones, namely:
or one can see the equivalent structure, which can also be considered alternatively as:
The discrete QOS, represented by the definitions (43) or (44), can be admittedly considered, without doubt, as finite dimensional representations of the original QOS, based on density function tags. This can be so, as both \( { \mathbf{z}_A } \) and \( { \mathbf{s}_A } \) discrete tags, essentially are elements of some VSS. Perhaps the representation (44), with tags belonging to the unit shell, corresponds to the most adequate of such discrete forms and, at the same time, the one which is more connected to the original infinite dimensional unit shell made by the collection of density functions.
Even if the choice to build up the problem is the DQOS, represented by Eq. (44), the fundamental QQSPR Eq. (40) can be transformed conveniently into the new row stochastic system:
simply by multiplying on the left by the inverse of the diagonal matrix (27), and using accordingly the transformed properties vector: \( { \mathbf{p} = \mathbf{D}^{-1}|\pi\rangle } \). The column stochastic transformation can be used straightforwardly too, and so it will not be discussed here anymore, as it has been exhaustively studied within several papers [59,100].
First-Order Fundamental QQSPR (FQQSPR) Equation
The analysis of the QQSPR problem can start with the first order or linear fundamental QQSPR equation, involving the core set, formed with the molecules of the associated DQOS, which are also linked with known values of some property.
One can write the first-order QQSPR fundamental equation in a compact matrix form:
Where the matrix Z is the already described symmetric QSM, \( { |\mathbf{p}\rangle } \) is the known core set property vector and \( { |\mathbf{x}\rangle } \) is a \( { (n\times 1) } \) vector, whose coefficients have to be evaluated.
The predictive power of such an equation is a priori null, because being the QSM: Z, by construction non-singular (otherwise two density functions will be exactly the same), then there always can be computed a QSM inverse: \( { \mathbf{Z}^{-\mathbf{1}} } \), obeying the usual relationships: \( { \mathbf{Z}^{-1}\mathbf{Z}=\mathbf{ZZ}^{-1} = \mathbf{I} } \), in such a way that the trivial result, defining the unknown coefficient vector:
will be always obtained within a core set scenario. Furthermore, one can retrieve the exact value of the property for any molecule of the core set QOS choosing the scalar products:
The QSM for diverse core sets has been used in a quite large set of prediction studies, in every case employing up-to-date statistical tools, the usual procedures currently available in classical QSPR studies. In the present study, the reader can find in the following sections new theoretical developments of the fundamental QQSPR equation prediction ability. However, a reminder of some simple linear algebra for the FQQSPR equation is needed first in order to understand the following arguments; therefore this will be described in the following sections.
Partitioning the FQQSPR Equation and the QSM Inverse.
Supposing now one can organize the QSM in the fundamental Eq. (40) in such a way that the last column and row correspond to a U-m, then, the unknown property element will be supposedly stored in the last position, the \( { (n+1) } \)th, of the vector \( { |\mathbf{p}\rangle } \) and will be symbolized by an a priori undefined parameter: p. With this in mind, one can design a partition of the QSM and the entire FQQSPR Eq. (40) in the following way:
Where, the \( { (n\times 1) } \) column vector \( { |\mathbf{z}\rangle } \) corresponds to the representation of the U-m in terms of the density tags of the core set and ? is the U-m self-similarity measure , which according to the simplified formalism of the expectation values can be defined by means of a simple overlap quantum similarity measure, as the Euclidean norm:
One can find the solution of the partitioned linear system by using the following symbols for the partitioned QSM inverse:
and one can evaluate the inverse elements for partitioned QSM matrices in the usual way.
Remarks on the Structure of the Fundamental QQSPR Equation
The following remarks relate to the result given by the fundamental QQSPR equation that was discussed in the previous section. Each of these remarks poses new problems that will be studied separately in subsequent sections.
Symmetrical Similarity Matrices.
In the first place, it must be said that the fundamental QQSAR equation has been usually presented in previous literature within the particular case where the basis and probe molecular quantum object tagged sets coincide, providing a square symmetric similarity matrix, and thus the equality: \( { \mathbf{A}=\mathbf{Z} } \), between the involved similarity matrices holds. This choice has the drawback that the fundamental QQSPR linear system becomes well defined, with a unique solution, whenever the similarity matrix is non-singular, which shall be the usual case, as far as no quantum object coincides with another within the quantum object set.
But even then, there is quite a wide range of solutions to overcome this apparent limitation. Among other procedures, one can use the symmetric similarity matrix as a source of molecular descriptors and afterwards employ them in classical statistical treatments. This choice, as was already commented, has been studied in many publications of our laboratory with success. In the same way, the similarity matrix can be transformed into a column or row stochastic matrix and, as a consequence, this form suggests several possibilities, which still are far from being exploited. Some analysis of the stochastic issue will be developed in a forthcoming section of this paper.
Origin of Hansch QSAR Models.
An interesting possibility of the symmetric square representation of the quantum similarity matrices corresponds to its potential to unveil the origin of one parameter classical QSAR models, such as those Hansch described some years ago. Indeed, under the equivalence of both the basis B and probe P quantum object sets, the FQQSPR linear equation corresponds to a set of N equations with the same number of unknowns, and can be rewritten as:
where there is no need to attach the similarity matrix elements to any specific quantum object set, as all of them are computed over a unique basis of density function tags. Considering the two terms at the end of the previous equation, it can be seen that the first one, with a diagonal value of the similarity matrix, is attached to a self-similarity measure z JJ , while the second term in cases of a not so strongly varying family of quantum objects, can be considered almost a constant, that is using:
the above equation takes the final form:
which has the required appearance to be considered as possessing a Hansch structure.
Besides this last deduction, it must be said that self-similarity measures of different kinds have been used to test this simple linear equation with quite a large series of quantum objects, yielding usually good results. Self-similarities can be sound substitutes of the classical Hansch analysis parameters. They constitute for co-generic molecular sets molecular descriptors with the property to be directly attached to a tri-dimensional molecular structure. Self-similarity measures vary slowly with conformational changes, so their values for the optimal geometry can be safely used, knowing that the magnitude of the descriptor will differ not very much from the one which is attached to the active conformation associated to the observable property.
Solutions of the Linear QQSPR Fundamental Equation
Equation (40) can be solved choosing the customary methods, as in the usual algorithms employed within the QSPR field [20]. They have been manipulated in this fashion, since its first description on a great deal of cases, as well as for a large variety of problems and subjects [26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,101]. This means that in all these studies the QQSPR fundamental equation has been solved with some algorithm, based on the least squares or similar technique [102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146].
However, the characteristic features of the QQSPR fundamental equation, and its definition within the VSS formalism, show that several alternative possibilities can be described, which will be studied next. Such non-classical solutions have in turn provided a collection of many new properties, concepts and application examples related to the tagged sets, VSS and IMP definitions.
Therefore, in order to exploit the QQSPR a plausible alternative to the principal components analysis will be proposed and then, the use of IMP and other techniques to obtain approximate solutions of the QQSPR fundamental equations will also be discussed.
Similarity Matrix Eigenvectors as Basis Sets to Construct the Solutions of the QQSPR Fundamental Equations.
Suppose we set the fundamental QQSPR Eq. (40) for a given problem. The secular equation of the involved quantum similarity matrix, Z, can be written as:
where: \( { \mathbf{C} = (\mathbf{c}_1 ,\mathbf{c}_2 ,\ldots ,\mathbf{c}_N) } \) is the matrix collection of the eigenvectors of the similarity matrix and: \( \Theta =\operatorname{Diag}(\theta_1;\theta_2;\ldots;\theta_N) \) is a diagonal matrix made by the ordered eigenvalues in descending order. The eigenvector matrix can be considered orthogonal, that is, the following property holds: \( { \mathbf{CC}^\mathrm{T}=\mathbf{C}^\mathrm{T} \mathbf{C} = \mathbf{I}_N } \), with the symbol: \( { \mathbf{C}^\mathrm{T} } \) indicating matrix transposition. The eigenvector associated to the greater eigenvalue has their entire elements positive definite, according to the Perron–Frobenius theorems [147]. The spectral decomposition:
can be used in Eq. (40), to obtain, after straightforward rearrangements, the equation:
so, renaming the set of scalars in Eq. (56) as:
then, the new equation could be written by means of a linear combination of the similarity matrix eigenvectors :
Equation (57) permits us to compute the new coefficients \( { |\gamma\rangle } \) in an obvious way, by using the orthogonal nature of the eigenvector matrix:
In fact, the original fundamental QQSPR equation coefficient vector can be obtained taking into account the spectral decomposition of the similarity matrix inverse, that is:
Therefore, Eq. (58) indicates that the coefficient vector, the solution of the previous Eq. (40), may be expressed by a linear combination of the eigenvectors of the similarity matrix too. Then, to every eigenvector, \( { \mathbf{c}_I } \), there is associated a well-defined scalar coefficient, ? I , which may be used as a reordering rule in order to obtain approximate solutions of the fundamental QQSPR equation. That is, suppose the eigenvectors are now ordered by the decreasing values of the set: \( { |\omega \rangle = \{\omega_I\} } \), then one can write:
where e is a given threshold splitting the vector construction in an approximate vector, \( { \mathbf{w}_{\text{a}} } \), and the remaining one, \( { \mathbf{w}_{\text{error}} } \), which can be interpreted or used as a residual error vector.
Stochastic Matrix Eigenvectors as Basis Sets to Construct the Solutions of theFundamental QQSPR Equations
Similar treatments can be designed for the QQSPR fundamental equation of the stochastic matrices, like the one employed for Eq. (50). The problem is that the stochastic matrix S is no longer symmetric and the attached eigensystem, apparently appears to be more laboriously solved, than in the case of the symmetric quantum similarity matrix Z. The problem has been already discussed in a general manner [59], so here only some simplified discussion will be given.
Suppose the secular equation, attached to the stochastic matrix S is written as:
substituting the matrix S by the expression of the row stochastic transformation as in Eq. (50), then:
which can readily be transformed by simple matrix manipulations and by using the square root of the diagonal matrix D, into the new secular equation:
which from here, calling: \( { \mathbf{A} = \mathbf{D}^{-\frac{1}{2}} \mathbf{ZD}^{-\frac{1}{2}} } \) and \( { \mathbf{Y}=\mathbf{D}^{\frac{1}{2}} \mathbf{X} } \), a new equivalent secular equation is readily made:
Equation (60) has the advantage that the matrix A is symmetric, hence the eigenvector matrix Y appears to be orthogonal: \( { \mathbf{Y}^\mathrm{T}\mathbf{Y} = \mathbf{YY}^\mathrm{T} = \mathbf{I} } \). Then, the sought eigenvectors of the stochastic matrix can be obtained by using the relationship between Yand X, that is:
The matrix D, in these circumstances acts as a metric with respect to the eigenvectors of S, as one can see that the orthogonality relationships:
hold, due to the orthogonality of the eigenvector matrix Y.
This proves finally that, in any case, Eq. (50) can be solved in the same way as previously commented for Eq. (40), simply by using the appropriate spectral decomposition:
Similarity Matrix IMP Decomposition in Order to Construct Approximate Solutionsof the QQSPR Fundamental Equations
QQSPR Fundamental Equation over VSS.
Going back to the QQSPR fundamental Eq. (40), now one can consider the positive definite nature of the elements, which appear to build the quantum similarity matrix. This can be expressed by means of the symbol: \( { \mathbf{Z}\ast > 0 } \) Footnote 2. The structure of the property vectors can also be taken into account. In the case when the following characteristic also holds: \( { |\pi\rangle\ast > 0 } \), for the involved QOS property or activity vector, then Eq. (40) can be associated to some linear transformation occurring on a VSS, for it can be written:
showing that a linear combination of vectors, belonging to a VSS, has to be brought into another vector belonging to a VSS.
This situation could only be generally achieved by using the condition: \( { \mathbf{w}\ast > 0 } \), which means that the linear system solution also shall belong to the VSS. Equ. (61) above becomes a constrained linear system of equations, since one is seeking solutions for which:
Approximate Restricted Solutions of Fundamental QQSPR Equation in VSS.
An approximate solution of the QQSPR fundamental equation can be obtained in the following way. As all the involved columns of the problem belong to a VSS, they can be decomposed by means of an IMP in terms of some IMP square powers of real matrices. If the treatment has to be more general the squared module of some complex matrices can be alternatively employed, but the treatment becomes slightly more difficult and the needed set of symbols heavier, so just real matrices will be supposed in this discussion. Therefore, owing to these considerations one can write:
So Eq. (61) can now be rewritten as:
suggesting an alternative approximate equation, which may be written in the following terms:
which has been obtained in turn, simply using the plausible approximation:
However, the approximate Eq. (63), suggests that the new linear system:
can now be solved, as it does not have to be submitted to any restriction at all, then:
and finally, the approximate solutions of the original system can be written as:
however this is sufficient to ensure:
The only problem, which now arises, is the existence of an inverse of the IMP square root of a non-singular matrix. Since the system (64) furnishes approximate solutions to the original problem (61) has to be found, there will be no major problem then to use approximate solutions in the least squares sense of the Eq. (64), as a way to obtain the approximate solution of the QQSPR fundamental equation, restricted to belonging to a VSS.
Convex Conditions Imposed on the Solution Vector of the QQSPR Fundamental Equations
Generating Vector Considerations.
The associated problem, to a form like Eq. (61), can also be solved, for instance, as in the well-known ASA fitting procedure [75,76,77,78,79,80,81,82,83]. That is, by using a convex condition on the solution vector: \( { K(\mathbf{w}) } \), with the additional meaning that the solution is now forced to belong not only to a VSS, but also to the unit shell.
If the solution of the linear Eq. (61) has to be found as an element of a VSS, \( { \mathbf{w}\ast > 0 } \), necessarily it has to be expressible as an IMP power of some generating real vector like: \( { \mathbf{w} = \mathbf{x}\ast \mathbf{x} } \). In choosing the convex conditions over the solution: \( { K(\mathbf{w}) } \), then the additional restriction is admitted to hold too:
However, this becomes the same as to consider that the generating vector \( { R(\mathbf{x}\to \mathbf{w}) } \) is normalized. Orthogonal transformations on the generating vector leaves the vector norm invariant, that is: whenever the condition (65) holds, and an orthogonal transformation U is performed on the generating vector, still the generating rule and the associated convex conditions apply: \( { R(\mathbf{Ux}\to {}^U\mathbf{w}) \wedge K({}^U\mathbf{w}) } \). Such an idea has been applied to obtain the ASA approximate density functions, using elementary Jacobi rotations [148] as a source of orthogonal transformations.
Stochastic Transformations.
Still more interesting appears the structure of the fundamental QQSPR equation, when the stochastic transform of the similarity matrix is considered. Equation (40) can thus be multiplied by the inverse of the diagonal matrix D on the right as defined in (27), providing:
which can be transformed into:
whenever it is considered that the following equalities hold:
Equation (66) can be also written as a linear combination of the columns of the stochastic matrix \( { \mathbf{S} = \{s_I\} } \):
Therefore, this is the same considering the s-shells of the vectors v and \( { |\pi\rangle } \) as being almost the same, as:
Thus, if the vector \( { |\pi\rangle } \) is transformed so as to become a unit shell element, this will be completely equivalent to applying the same transformation into the transformed unknown vector v. Therefore, the following implications are straightforwardly deduced:
Demonstrating that in the stochastic QQSPR fundamental Eq. (66) case, the solution contained within a given VSS amounts to the same as obtaining a convex combination of the stochastic matrix columns.
Stochastic QQSPR Least Squares Solution via Jacobi Rotations.
After all the previous discussions, there appears to be another possibility which has remained unexplored. Starting from the transformed Eq. (66), with the appropriate definitions (67) in mind, one can seek an approximate solution of the stochastic equation in the least squares sense, defining the quadratic error function by means of the difference vector:
whose Euclidean norm furnishes the quadratic error function, expressible in terms of a scalar product or the inward product sum:
One easily arrives at the quadratic form:
however, the quadratic error function optimization has to be carried out preserving the condition of convexity \( { K(\mathbf{v}) } \) on the solution vector, otherwise one risks obtaining solutions that do not belong to the unit shell. In order to obtain an appropriate algorithm to perform this task, the following considerations can be taken into account.
The vector v can be expressed as an IMP of an auxiliary vector a, that is:
then the belonging of v to the unit shell is equivalent to the Euclidean normalization of a:
The quadratic error function (68) can be expressed in terms of the auxiliary vector:
where the symbol: \( { \mathbf{a}^{[\mathbf{2}]} = \mathbf{v} = \mathbf{a} \ast \mathbf{a} } \) has been used. Also, employing to simplify the notation the following conventional symbols:
so Eq. (70) can be explicitly written as:
Starting with an approximate normalized auxiliary vector, orthogonal transformations can be performed, preserving the norm, thus keeping the condition \( { K(\mathbf{v}) = K(\mathbf{a}^{[2]}) } \) constant along the optimization of Eq. (71). Orthogonal transformations can be chosen as elementary Jacobi rotations [148], which at every application over the vector a, change two chosen elements \( { \{a_P;a_Q\} } \) into a pair of new rotated ones \( { \{a_P^R;a_Q^R\} } \), according to the well-known algorithm:
where \( { \{c,s\} } \) are the cosine and the sine of the rotation, with the additional obvious relationship: \( { c^2+s^2=1 } \).
Over the generating vector coefficients in Eq. (69) it is easy to apply the EJR represented by Eq. (72), and then, the variation of the quadratic error \( { \delta\varepsilon^{(2)} } \), with respect to the active pair of elements \( { \{a_P;a_Q\} } \) may be easily expressed.
Taking also into account that the quadratic elements, for example, will transform and yield variations like:
A Jacobi rotation as shown in the expression (72) will produce a variation in the quadratic error (71) in the chosen rotated elements, when taking also into account the symmetric nature of the matrix H, of the form:
so, also needed are the quartic variations of the auxiliary vector elements, which can be easily computed as in the second-order case.
Substituting such variations into the corresponding equation and collecting terms one finally arrives at a quartic polynomial on the rotation sine:
where the parameters \( { \{E_{IJ}\} } \), appearing in Eq. (73), are described as follows:
using the following auxiliary terms:
and
The optimal sine can be chosen with the null gradient condition \( { \mskip2mu\mathrm{d}\delta\varepsilon^{(2)}/\mskip2mu\mathrm{d} s = 0 } \), taking into account that: \( { s/c=t } \) and that: \( { \mskip2mu\mathrm{d} c / \mskip2mu\mathrm{d} s =-t } \), then:
holds with the auxiliary definitions:
The best Jacobi rotation angle is found solving the quadratic polynomial equation in the EJR tangent {t}, appearing in expression (74). The optimization can be conducted through an iterative procedure, until the global variation of Jacobi rotation angles or the quadratic error integral function become negligible. The interested reader is conducted to the references [78,79,80] for more details, where a complete account of all the Jacobi rotation techniques can be found. A simplified algorithm can be also used and it will be briefly commented upon here. The procedure is based on the fact that sine and cosine can be written in function of the rotation angle:
and for small angles it is only necessary to use, up to second order:
so Eq. (73) can be transformed into a second-order polynomial in the rotation angle:
which submitted to the extremum conditions yields:
Non-Linear Terms and ExtendedWave Functions
Sobolev Spaces
From early times, quantum mechanics has been emphasizing not only the role of well-behaved wave functions, but also the relevance of their gradients and Laplacian forms. The reason for such requirements, necessarily holding on the current wave functions, can be simply connected to the presence, in the Schrödinger equation set up, of a second-order derivative, the result of a Laplace operator application, associated to the quantum system kinetic energy term, see for example the references [28,149].
Usually, the adequate quantum mechanical behavior of the wave function is focused, among other simple and obvious mathematical features, to the compulsive property that wave functions have to be square summable. In some reference books such a property has been promoted to the category of a postulate [150] and in the very early development times of quantum mechanics [98] has been interpreted by Born as the fact that the square module of the wave function can be associated to a probability density function. It was von Neumann [97], who related such properties, among other crucial quantum mechanical theoretical elements, with the mathematical structures of Hilbert–Banach spaces [94,151,152]. More recently, Landau and Lifshitz [153] described the role of the wave function gradient as a descriptor of infinitesimal translations and rotations. These authors settled as well the use of the square module of the wave functions gradient, in order to obtain an alternative kinetic energy expectation value expression, more likely related to the statistical formalism than the Laplace operator form.
On the other hand no utilization has been reported of the so-called Sobolev spaces [154] in applied quantum mechanics, at least to our knowledge. Curiously enough, Sobolev spaces where defined as early as 1938, and apparently have been of practical use in some remotely related theoretical landscape, associated to generalized relativity applications [155]. It was not until recently that Sobolev spaces were proposed by us as a vehicle to take into account the role of the wave function squared module:\( { |\Psi|^2 } \), as well as to make simultaneously relevant the presence of the wave function gradient squared module:\( { |\nabla \Psi|^2 } \) in an extended density function. In all this previous work both terms were also presented as forming part of a new quantum mechanical composite norm. In this way, the classical quantum mechanical Banach space has been transformed into a Sobolev space structure [14,156,157], without losing generality, but gaining flexibility instead.
Sobolev spaces can be defined in several ways, leading all of them to simple forms, ready to be used in reinterpreting the approximate solution of the Schrödinger equation and prone to be included with immediate applications, such as those found among the references [156,157]. They can be constructed in such a way as possessing extended forms even more complex, in order to be used to include arbitrary non-linear terms in the same equation [14]. This can be understood by recognizing the fact that the Banach space can be considered the limiting simplified form of a quite large collection of Sobolev spaces.
Quantum Mechanical Hilbert and Banach Spaces.
In order to present the Sobolev spaces step by step, the simplest formalism will be defined first, and other extended possibilities will be described later on. To achieve this objective, suppose a quantum mechanical wave function Hilbert space, which can formally be described as:
where the symbol r, used as the wave function variables, shall be considered as a vector, containing all the necessary particle position coordinates as its components. The number of particles is shortly noted with the dimension P of the coordinates vector space. The wave function elements of the Hilbert space (75), possess as a sine qua non condition, the following well-known property about the existence of a positive definite density function, which is remembered here, just to present the notation that will be hereafter employed:
Besides, the density function attached to every wave function, as proposed in Eq. (76), can be seen as belonging to a Hilbert semispace, \( { H_\infty(\mathbf{R}^+) } \), where all function values and coefficients are strictly allowed to be positive real numbers only. That is: in the same way as in the Hilbert space (75), one can write the corresponding formal definition for the density function semispace:
where the dimension of the coordinates vector space, \( { V_P(\mathbf{R}) } \), containing the density function variables, has the same meaning as in definition (75). Moreover, the Hilbertspace (75) is a Banach space, as all of their elements shall compulsively fulfill the normalization condition:
which obviously amounts to the same as imposing on every element of the Hilbert semispace (77), a convexity condition:
Gradient of the Wave Function.
As was previously commented, the whole quantum mechanical Hilbert space elements shall present other existence properties, mainly related to their derivatives, for instance:
where the nabla operator ? refers to the gradient with respect to the vector coordinates r. That is, formally it can be also written: \( { \nabla \Psi \equiv \partial \Psi/\partial \mathbf{r} } \). The ordering of the resultant gradient vector elements can be somewhat arbitrary; this means that they can be adapted to the structure of the operating mathematical context. In addition, associated to this mentioned component ordering, the resultant gradient vector can be considered to belong to some appropriate Cartesian product of the initial Hilbert space (75), which can be generally defined and noted in a simplified fashion as:
because in any ordering case, the resultant gradient vectors will depend on the particle number P. At the same time, the gradients of type (80), can also be easily associated to squared gradient modules, which shall belong to some Hilbert semispace, very similar to the one defined in Eq. (77):
where the positive definite function \( { \kappa(\mathbf{r}) } \), will produce, when integrated, twice the quantum mechanical kinetic energy expectation value \( { \langle K\rangle } \) of the attached system:
which as is well known, can be described alternatively like the classical quantum mechanical expectation value of the Laplace operator :
just employing Green's theorem [158].
The interesting thing to be said now consists in proposing some sentences on the nature of the integrals (83) and (84), which are real and positive definite as kinetic energy shall be, either classically speaking or quantum mechanically, thus providing the integral (83) with a well-defined structure, capable of being interpreted as a norm. It should also be noted that the imaginary unit, usually employed in the quantum mechanical definition of linear momentum does not need to be used here in front of the nabla operator. The reason can be found in the fact that the imaginary unit has no active role in the above definitions, unless a Hermitian matrix representation is needed for the ? operator. Such an imaginary factor has to be present in such a Hermitian representation case, because the matrix associated to the bare nabla operator is Skew-Hermitian, that is:
a property which can be easily interpreted as a consequence of the application of Green's theorem again.
The Simplest Sobolev Space.
In any case, the existence of the wave function norm (78) and the subsequentconvexity condition (79), can both be recognized as the parallel properties holding for the gradient of the wave function, and corresponding to Eq. (83), which proves collaterally the positive definite nature of the quantum mechanical kinetic energy expectation value. Thus, if the sequence of equations from (75) up to (83) must hold simultaneously, just to obtain a coherent mathematical structure within the quantum mechanical framework, it is feasible to consider that both Banach spaces (75) and (82), can be supposedly forming a composite norm, in the way of the following definition present within the equation shown below. Simplifying the wave function notation from the variable dependence, in order to ease the form of the subsequent equations, one can define the following norm:
Such a composition provides the first definition of the simplest element among the collection of all possible Sobolev spaces, which can be connected to quantum theory. Thus, it can be assumed from now on that the most adequate quantum mechanical wave function space structure is a Sobolev space .
Sobolev Spaces.
The notation for the norm (85) will be made immediately obvious, by means of defining a general Sobolev space norm as:
The usual Hilbert–Banach space norm can be retrieved from definition (86), simply supposing that the null power of the gradient operator can be substituted by the identity: \( { \nabla ^0\equiv I } \), and using afterwards: \( { \alpha =1\wedge \beta =0 } \). In addition, the earlier Sobolev space norm, simplified as in Eq. (85), is also found employing Eq. (86), but choosing: \( { \alpha =\beta =1 } \).
However, although Eq. (86) contains the classical quantum mechanical Sobolev ?-norm, it implicitly possesses the restriction consisting in that both wave function and gradient norm powers shall be the same in any circumstance. Consequently, they cannot be monitored as independent terms in the norm definition. An appropriate choice to avoid this situation may be described with the more general formulation:
Therefore, Eq. (87) will transform into expression (86), whenever: \( { \alpha =\gamma } \). In order to avoid further interpretation problems, the squared module of the nabla powers has to be considered a contraction operation; or has to be considered a scalar product of the corresponding matrix elements, represented by the result of the operation \( { \nabla^b\Psi } \):
Nested Summation Symbols.
This last remark, represented by the Eq. (88), can be alternatively written in a very elegant manner employing the definition of an inward matrix product (IMP), already discussed.
Taking the IMP definition into account, then in Eqs. (86) and (87) it can be understood that the present square modules are computed over the resultant wave function derivative hypermatrices as:
where the symbol \( { \langle\enskip\rangle } \), associated to any matrix, means a sum of the whole matrix elements, for instance:
which constitutes a definition possessing obvious generalization possibilities, within any kind of hypermatrix structure.
This generalization power can be easily seen, taking into account the nested summation symbol (NSS) formalism, which was developed several years ago, see references [159,160] for example. Then, using NSS, the expression of the total sum of the elements of an arbitrary hypermatrix can be generally written without any further problem. A NSS is a symbolic device, which has a linear operator nature, and in this way resumes the presence of an undefined number of nested sums and corresponds to an easily programmable algorithm, which generalizes in practice an indefinite number of do loops. In turn, a NSS acts over any kind of complex expression, bearing all the involved indices present within the sums, that is:
where by the index vector i it is understood: \( \mathbf{i} = (i_1;i_2;\ldots; i_N) \). Thus, if by the definition the following subindex structure is assumed:
by which is represented any \( { (n_1 \times n_2 \times \ldots n_N) } \)-dimensional hypermatrix element, then the symbolic device associated to the total summation of the elements, particularly defined in Eq. (89), can be generally described by means of the compact NSS expression:
Extended Wave and Density Functions
Sobolev spaces appear, after the previous discussion, as a very general kind of extended Hilbert–Banach spaces, which within the quantum mechanical framework are able to put into a unique statement the nature of both the wave function and its gradient. Alternatively, they can produce completely general structures, somehow involving usual quantum mechanical operators, attachable to any system observable. It is a matter of straightforward analysis to translate the subjacent Sobolev mathematical structure into the Hilbert space elements themselves, producing a new breed of spaces, which can be obviously called Hilbert–Sobolev spaces Footnote 3. As has been done before, within the previous description of Sobolev spaces, the extension of the wave function and the possible application of the resultant formal structure will be here gradually discussed.
Extended Wave Functions.
By a ?-extended wave function \( { |\Phi\rangle } \) has been understood a composite column vector, or alternatively a diagonal matrix, if one prefers, whose elements are the original wave function ? and its gradient \( { \nabla\Psi } \). That is:
The column vector form, as will be discussed later on, better represents some applications and the mathematical manipulations one can perform over them; while in other cases the diagonal matrix structure produces more elegant expressions and it is easier to deal with. However, both choices provide equivalent results. This is so because the proposed representations constitute the elements of a pair of isomorphic vector spaces.
The previous discussion on Sobolev spaces permits us to define the extended wave function within a general Hermitian operator scheme, simply as:
The definition of Eq. (91) can be called an O-extended wave function . For example, the quantum mechanical complementary definition of the ?-extended wave function (90) can be easily written employing the position vector r, that is:
producing an r-extended wave function accordingly.
By inspection of the adopted structure until now, extended wave functions can be also considered as the result of applying some adequate operator over the original Schrödinger wave function. For instance, within the already-mentioned diagonal formalism of the O-extended wave function (91), defining the diagonal operator:
where \( { \textbf{I} } \) is the unit operator, it will be sufficient to see that:
The same can be said if the corresponding vector operator is constructed by means of the vector structure:
which, upon application over the original scalar wave function form, permits us to alternatively obtain the isomorphic vector picture of the diagonal expression (93).
Energy Expectation Values.
Returning to the ?-extended wave function in Eq. (90), it is easy to see how the energy expectation value of the associated Schrödinger equation can be expressed, without loosing any information, when writing the final form it takes. For this purpose, the appropriate Hamilton operator, H, can be structured by means of a diagonal form, as:
where the symbol: U, corresponds to the potential energy operator and I is just a unit operator built to fit the adequate dimensions of the extended wave function (90) gradient part. Using Eq. (94) and the ?-extended wave function (90) in the appropriate way, it is immediate to write:
In the same way, whenever the normalization of the wave function holds: \( { \langle\Psi|\Psi\rangle =1 } \), the ?-extended wave function (90) can be manipulated in order to obtain a positive definite norm like:
which corresponds to the same result as the one provided by the norm obtained in the definition of the simplest Sobolev space, as presented first in Eq. (85). Obviously, such a Sobolev space can be interpreted as a composite Hilbert space, whose elements are defined by the ?-extended wave function (90). That is, employing the Cartesian product of the Hilbert spaces (75) and (81), upon reordering the ordered pair in the form of column vector:
Extended Density Functions.
The Hilbert semispace corresponding to the Hilbert space (97), can be supposedly formed by the total density functions and computed by using the trace of the extended wave function tensor product, for instance:
that is: as the superposition of the electronic density \( { \rho(\mathbf{r}) } \) and the kinetic energy density \( { \kappa(\mathbf{r}) = |\nabla\Psi|^2 } \). Such a result appears to be consistent with the previous definitions, as already expressed in Eqs. (85) and (96). An alternativedefinition of the total density (98) can be also obtained, employing the already defined IMP, in association with the total sum of elements of a vector, as previously given in Eq. (89) and generalized afterwards:
Such superposition of density functions, producing the total density function \( { \tau(\mathbf{r}) } \), can be obviously named as a ?-extended density function .
It must be noted now, that the total density function, as defined in Eqs. (98) or (99), possesses the statistical interpretation of observing, within a space infinitesimal volume element, both the position of the associated system of particles or the same system within a corresponding related infinitesimal kinetic energy range. The conjunction or, is a consequence of the obtained statistical expressions, where both densities are summed up, and it is in agreement with Heisenberg's uncertainty principle , which will forbid the practical use of the product of both distributions, position and momentum not being simultaneously observable.
Quantum Self-Similarity Measures and Non-linear Schrödinger Equation.
The previous experience, crystallized in definitions (90) and (91), about the extended wave functions and the details of their subsequent use in the energy definition (95), as well as the construction of the extended density function in Eqs. (98) or (99) formalisms, shows a plausible way to generalize the presented wave function extensions. This effort can be employed as a way to try, afterwards, to obtain information on the possible utility of such extended general wave function forms.
In the same way as the generalization of Sobolev spaces, starting from the simplest form (85), the O-extended expression (91) of the wave function can be generalized accordingly.
In order to do so, a systematic exposition will be followed, in the same way as has been previously done. Thus, the first new breed of extended wave functions will be defined by means of the vector, or diagonal, form:
As the second element of the vector (100), is simply the electronic density function, such a vector can be written equivalently as:
and the corresponding possible energy expression could be written in turn, employing a Hamilton operator and following the previous experience as presented in Eq. (94) in the form:
where the first and the last non null elements are the same as shown before in Eq. (94), and besides a is an arbitrary real parameter. Thus, the energy expectation value equivalent to the expression (95), employing in theextended wave function (101) the substitution \( { \Omega =\nabla } \), gives:
So, calling the classical Schrödinger energy (95):
and owing to the fact that the second term in the expectation value (103), can be manipulated as follows:
then, one can simply write:
The nature of the integral (104) is well known in the theoretical formulation and definitions of quantum similarity measures (QSM), corresponding to the so-called quantum self-similarity measure (QSSM) associated to the density function ?. As is evident upon inspecting the QSSM appearing in Eq. (104), the integral also corresponds to a scalar product of the density function by itself. That is: a simple Euclidean norm within the associated Hilbert semispace, containing ?. In addition, it can be quantum mechanically interpreted as the expectation value of the density function over itself. Finally, the integral is also closely related to a relativistic correction appearing in the definition of the Breit Hamiltonian : the term named spin-spin contact [161,162,163], although in the present form (104) the two-electron Dirac d-function is absent. The characteristic non-trivial features of this kind of QSSM integral, as appear in this particular formalism, and more precisely with respect of the spin part of the wave function, have been deeply analyzed in two separate papers [87,164].
Moreover, Eqs. (101) up to (105), tell that, in fact, the new extended wave function produces an energy expectation value, which can be seen to be in correspondence with the non-linear Schrödinger equation.
Upon inspecting definition (100), or Eq. (101), the corresponding extended density function can be deduced, employing the same technique as the one that was previously used in Eq. (99):
So, the total density now can be written as the ?-extended density function (99), with an extra term made of the squared electronic density:
Finally, the Sobolev norm of the extended wave function (101) will be easily obtained by integrating the extended density function (106):
the QSSM integral z, defined in Eq. (104), can be also considered as a norm, associated by construction, to the elements of a Hilbert semispace , then this fact assures the positive definition of the integral (107). It is not difficult to associate the norm in Eq. (107), with a Sobalev normof type (87), with the parameters chosen accordingly: \( { \|\Psi\|_2^{1,1} } \).
Thus, a naïve generalization of the idea underlying the ?-extended wave function definition has revealed itself as a powerful tool, which permits the formal description of the non-linear Schrödinger equation. Such formalism allows producing another kind of Hilbert–Sobolev space, and at the same time, within their integral steps, finally puts into evidence the connection of Sobolev spaces and extended wave functions with the concept of QSSM.
Expectation Values Within Extended Density Functions Framework
Landau and Lifshitz proposed the interpretation of expectation values in a statistical formulation, instead of the usual quantum mechanical form. This was anticipated somewhere in the already quoted volume of reference [153], published within a series dedicated to studying the mechanics of particle systems. The same point of view was also masterly described and adopted, later on, by McWeeny and Sutcliffe in the book of reference [165]. In the present paper, employing the concepts associated to the extended wave functions, it will be shown that a similar possibility as the one mentioned in these previous references can be exactly deduced. The difference with the above-mentioned sources consists of the fact that, in the present way, one only needs to base the arguments on the structure of the already described Hilbert–Sobolev spaces. Some related point of view has been found in the same direction, precluding this property, when the deduction of the energy expectation values has been discussed, as can be noticed when observing Eqs. (95) and (103). According to this, the purpose of this section is to deduce a general Hilbert–Sobolev formalism for the expectation values associated to the extended wave functions and provide an application example.
Statistical Form of Expectation Values in the Extended Wave Function Formalism.
One can deduce the general composition of an extended density function, corresponding to the extended function (91). For this purpose, an appropriate operator shall be defined. It has to be able to act over the extended wave function structure. Accordingly, it is sufficient to take into account that a diagonal-like operator can be constructed in the following way:
where, in order to be applied along the appropriate extended wave function elements, the involved operators G and ? themselves have to possess an adequate structure. When, within the global expectation value expression, the O-extended wave function is used quantum mechanically over the operator (108), in order to obtain the corresponding equation, then owing to the properties of diagonal matrices, both the wave function and the diagonal operator can be manipulated, in the forthcoming manner, employing obvious notation and symbols, to arrive towards a statistical formulation final form:
Now it must be taken into account that the external summation symbol employed in the above equation, has to be taken, when appropriate, as an integration procedure.
Energy Expectation Value of a Set of Interacting Quantum Objects.
Among the possible uses of the present formalism, it seems worthwhile to consider some theoretical arrangement associated to a previous discussion made by Huzinaga and co-workers [166,167]. This procedure is related to the model potential method, proposed by Bonifacic and Huzinaga [168] in order to study the optimal valence AO, transforming the core electron structure into an electrostatic potential. In the following discussion the structure and final form will be presented, which can take the total energy, when the problem of several interacting quantum objects is studied in a somehow approximate way from the point of view of the wave function. In order to perform such a study, one can suppose known a set of quantum objects [25], whose Hamiltonian operators in this case can be considered constructed with a diagonal structure, similar to the one described in Eq. (102).
However, the second diagonal term has to be transformed necessarily into a new operator and described with the appropriate construction rule, as follows:
This operator is in some way the reciprocal mirror image of the well-known unit operator:
In the last definition (111) as well as in the operator previously defined in Eq. (110), a logical Kronecker symbol has been utilized [159,160]. Considering the definition of the unit operator (111), the logical Kronecker symbols appear obviously structured, adopting a self-explanatory description. The definition, for instance, can be made clearer, considering a logical expression ? taken as the Kronecker symbol argument, and then its resultant value can be generally described by means of the logical content of the possible issues of such an argument, that is:
The operator \( { \mathbf{1} } \), the unity operator as used in Eq. (110), means the multiplicative unit of the IMP, that is: \( \mathbf{1}=\{\mathbf{1}_{ij} =1\} \).
Considering a as a parameter to be adjusted, according to the nature of the problem, then the Hamiltonian could be written in this case as:
where U is a scalar potential operator, defined by means of the expression:
with the sum encompassing all the involved quantum objects. By the symbol V I is understood an attractive potential; while by R I , the repulsion terms can be somehow described. Both operator terms have to be associated in turn to the whole set of particles, constituting the Ith quantum object. For instance: when dealing with atoms and molecules, these operators could be associated to the nuclear attraction operator and to the Coulomb-exchange operator terms, respectively.
Then, in this problem the appropriate extended wave function can be taken as the ?-extended form of Eq. (101), that is:
where the wave and the density functions present in the extended function (113) shall be taken as vectors, having as elements the wave and density functions of every quantum object in the considered set, respectively.
Thus, the expectation value of the Hamiltonian (112) under the extended wave function (113), can be easily written, using the technique of Eq. (109) as:
in the above expression the first term corresponds to the potential energy of the objects plus their interactions; the second term can be associated to the expectation value of the projection operator over each quantum object except itself, and the role of this part of the expectation value is intended to prevent the collapsing tendency of the particles, belonging to each separated quantum object, towards a unique system; finally, the third term corresponds to the global kinetic energy obtained in a way such as the quantum objects were non-interacting.
The second element of the expectation value (114) can be also easily interpreted as a sum of the overlap QSM [169] between pairs of quantum objects. In this sense, one can observe Huzinaga's treatment as a procedure, taking into account non-linear terms in the approximate solution of the Schrödinger equation.
Quantum Similarity Measures in Extended Hilbert–Sobolev Spaces
QSM in Hilbert semispaces have been studied from the theoretical point of view as well as considering the potential applications of quantum similarity over quantum objects. In this paragraph, the structure of QSM over extended wave and density functions will be analyzed. Before proceeding towards such an analysis, it must be said that, as QSM are essentially defined over density functions, they can be constructed even in the Hilbert–Sobolev spaces framework, provided that the extended density function is known. This is so, because, whenever a total density function can be well defined, like the one present in Eq. (98), for instance, then the construction of any similarity measure can also be put forward. Such a general possibility was analyzed in a particular way several years ago [170], when discussing the extension of the QS concepts into partition functions. Statistical mechanics partition functions can be obviously observed as probability distributions and, thus, they can be considered as elements belonging to a characteristic vector set: a Boltzmann semispace , for example. From such a fact they can be used in the general definitions of QSM, as any probability density function can be used for the same purpose.
The most usual way to produce a QSM, corresponds to the integral constructed as:
Where in Eq. (115), \( { \{\rho_I(\mathbf{r}_1);\rho_J(\mathbf{r}_2)\} } \) is a pair of homogeneous order density functions, and \( { \Omega(\mathbf{r}_1;\mathbf{r}_2) } \) is a positive definite operator. The attached properties of the set of integrands ensure that in any case the values of the QSM, defined such as in Eq. (115), will produce a positive real element. For the present purposes, the integral form (115) is sufficient. The already mentioned overlap QSM, which appears in the building up of energy expectation values within non-linear Schrödinger equations , such as those in expressions (104) and (114), are overlap-like QSM, and can be deduced from the QSM Eq. (115), by simply using a Dirac delta function as operator, that is: \( { \Omega(\mathbf{r}_1;\mathbf{r}_2) = \delta(\mathbf{r}_1 -\mathbf{r}_2) } \).
The definition of kinetic energy density and other possible density kinds, deducible from the O-extended wave function concepts, as discussed earlier, opens the way to produce QSM using the integral (115), upon substitution of the density function pairs by the appropriate extended density function.
So it seems now clear that the QSM integral form, as described in Eq. (115), can be used as it is for extended density functions, just substituting the usual electronic density by the corresponding expression in terms of the chosen extended density functions. Here, the interesting new feature consists of the emergence of QSM integrals, associated to density functions of different origin. For instance, suppose the ?-extended density function as defined in Eq. (98): the total density defined there, associated to a pair of quantum objects produces a QSM, which can be written in terms of four hybrid QSM integrals as:
The first term being exactly the one in Eq. (115), and the last one being a QSM over kinetic energy density distributions, the central terms corresponding to hybrid QSM between electronic and kinetic density functions. Total density QSM integrals of any kind still are waiting to be practically employed in a systematic way. It is important to consider them as potentially interesting quantum object descriptors. Because of their flexible generality, the extended density functions may provide new insights and refinements within QSPR models. In the next sections of this work a discussion of several possible uses of the extended densities technique will be discussed.
Fundamental Quantum QSPR (QQSPR) Equation in Sobolev Spaces
From the previous considerations and the ad hoc definitions, it can be said, without doubt, that the general structure of Sobolev spaces can be easily associated to the extended Hilbert wave functions, producing a new mathematical structure, which has been named a Hilbert–Sobolev space.
Hilbert–Sobolev spaces have been used too to obtain up to now, with the appropriate definition of the Hamilton operator, adapted to every circumstance, the corresponding energy expectation values. Elementary reasoning permits us to arrive towards the conclusion that the expectation values of a given observable can be obtained in a similar generalized manner and within the formalism, which is the most valuable finding, perhaps, of the present discussion: the possibility to write the expectation value expressions using a statistical method.
Thus, if some observable O, has attached the Hermitian operator T, and the associated quantum object extended density function is t, then the associated quantum mechanical expectation value, \( { \langle\vartheta\rangle } \), can be written, according to the previous considerations as:
In QQSPR reasoning, the expression above can be further arranged in the following way. The Hermitian operator T is usually not known, but it can be expressed as the product of a still unknown operator W by a known one O, associated at the same time with a positive definite property, that is:
Because O is chosen as having a positive definite form, one is always assured that knowing T, then W can be obtained in turn as:
However, in practice T and W are unknown, and in the expectation value formalism there is always the need to obtain W in an approximate way, employing a least squares procedure. This can be easily done whenever a set of compatible density functions, connected to a certain quantum object set as quantum object tags, is already known: \( { T=\{\tau_I\} } \) and can be used in order to express the operator W as a superposition such that:
Then, the expectation value (116) becomes expressible as:
being the resultant QSM integrals, defined in the usual way in Eq. (115). The well-described procedure to obtain the coefficients: \( { \mathbf{w} = \{w_I\} } \) is the least-squares technique, or anyone of the existing variants, as previously described, by means of an alternative method based on IMP reasoning.
The first step is, in any case, to proceed with the construction of a linear system of equations, whose solution is the coefficient vector w. To obtain such a linear system it is necessary first to know, for a given quantum object set, a set of properties: \( { \{\pi_I\} } \), which can be associated to the corresponding expectation values: \( { \{\vartheta_I\} } \). Then, employing the quantum objects density functions, which in particular will coincide with the set \( { T=\{\tau_I\} } \), used to construct the unknown part of the operator, although not necessarily both density function sets shall be the same, it is possible to write:
so collecting in a vector like \( { |\pi\rangle } \), the property values, and defining the integral matrix elements by means of:
it is easy to transform the system into a matrix equation in the form of:
The solution of the above system will make known an approximate form of the implied operator and, in this way, provides a possible path to be followed in order to obtain estimates of the property values of any unknown quantum object, just as in the classical QSPR model procedures.
However, the present kind of quantitative structure (represented by the QSM)-property model is completely based on quantum mechanical propositions. More than this, there are no other suppositions than the usual ones, associated to density function algebra and quantum mechanical basic mathematical background. Hence, the results obtained through the linear system of Eqs. (119), do not depend on user choice, but rely directly on theoretical grounds and because of this are statistically unbiased. Equation (119) can be properly called the fundamental QQSPR equation . Moreover, the models obtained in this way can be interpreted in the light of the quantum mechanical expectation value concept. By this simple fact, contrary to the classical QSPR modeling results, they can be associated to a causal relationship relying on quantum object properties and QSM.
Due to the unavoidable presence of the QSM matrix (118) into the fundamentalQQSPR equation, the columns of such a matrix play a fundamental role in the discrete representation of quantum objects. Consequently, the columns of the QSM matrix, involving a given quantum object density function, interacting with the whole basis set of density functions employed to represent the unknown operator W, can be safely considered as natural quantum mechanical discrete descriptors of the associated quantum object.
Non-Linear Terms in QQSPR Models
The usual relationships between structure and properties sometimes needs the presence of non-linear terms. Non-linear terms are needed in order to represent accurately the property as a function of the structural descriptors.
The fundamental QQSPR equation can be deduced to introduce in a natural way these terms, if needed. To see this, it is only necessary to think about the easy path, which was used to introduce the non-linear terms in the Schrödinger equation, just by using simple considerations, associated to the structure of the Hilbert–Sobolev spaces.
Suppose that the extended wave function (106) is employed, along with the corresponding extended density functions. In this case, although the Eq. (117), producing the unknown operator expression, can be supposedly set in the same manner as has been proposed in the usual framework, the kinetic energy distribution as well as the non-linear density terms can be employed separately. That is, the unknown operator can be written now as:
Such an approach will produce a set of linear equations, with an extended number of parameters, but also a matrix representation of the unknown operator with added dimensions. The matrix elements, involving squared density functions, are candidates to be interpreted as the quantum representatives of the possible presence of non-linear terms in the fundamental QQSPR equation.
Non-linearity can be introduced in several alternative ways, due to the flexibility promoted by the ideas around the Hilbert–Sobolev concepts. For example, within the QSM matrix definition (118), the operator O can be seen as formed by the expression:
which is assured to be positive definite whenever the exponent a possesses positive values. Convergence can be assured whenever: \( { a\in(0,1) } \).
In fact, such an expansion can be generalized by using a set of positive definite coefficients \( { A=\{a_I\} } \), such that:
In such a general case, the QSM matrix elements will be written as a superposition of terms like:
Where, in the last line of Eq. (122), it is easy to observe the overlap similarity integrals as the zero-th order term, the triple density similarity integrals as the second element constituting the first-order term, and finally, in the second-order term, the quadruple density integrals appear. Such integrals can be readily defined by means of the expression, chosen among other possible definitions, for example, as:
This result is still more obvious if the following operator structure is employed: upon substituting in Eq. (121) the density function by a convex superposition like the one in Eq. (120), which to obtain simpler expressions will be written as a convex superposition like:
being the coefficients \( { \{\omega_A\} } \) such that: \( \forall A\colon\omega_A\in \mathbf{R}^+\wedge \sum\limits_A\omega_A = 1 \).
In this case, the QSM integral (122) will take the following form:
It can be seen that quadratic or higher order terms can naturally appear in the structure of the fundamental QQSPR equation in this way.
Non-Linear Terms and Variational Approach in Quantum QSPR
Fundamental QQSPR Equation in \( { (N \times M) } \) Similarity Matrix Spaces.
Suppose a quantum object basis set B composed by M quantum systems, whose homogeneous density functions, acting as quantum object tags, are known: \( { B=\big\{\rho_I^B|I=1,M\big\} } \). Suppose also that a probe quantum object set P is well defined and composed by N quantum systems, which have also known density tags: \( { \big\{\rho_I^P\big\} } \), and at least is also known a set of property values: \( { \{p_J\} } \) attached to every quantum object of the set; in this manner: \( { P=\big\{\rho_J^P \wedge p_J|J=1,N\big\} } \).
A general operator O can be associated to the expectation value computation of the observable property p, in such a way that, knowing the appropriate quantum state density function tag ? for a given quantum system, such a quantum object observable property can be evaluated in general by using the integral form:
where D is an appropriate integration domain, where the density and operator variables are defined.
Being the operator O, in principle, after the adoption of quantum mechanical rules, a Hermitian operator, without loss of generality can be supposedly decomposed into a product of two commutative operators:
the operator T being a known chosen positive definite one, the remnant Hermitian operator is thus defined as:
Using Eq. (123) and the operator composition shown in Eq. (124), then it can be formally written:
suggesting that the operator W could be approximately obtained, even in the case that it is unknown, due to the nature of the observable attached to the property.
In the case, most usual in QQSPR framework, that an approximate construction of the operator W is needed, if an appropriate quantum object set density function tag set, acting as a basis set, B say, is known, as stated at the beginning, that is: \( { B=\big\{\rho_I^B | I=1,M\big\} } \), then the operator W can be written within a first-order linear approach as:
so upon substituting this approximate first-order linear expression into the expectation value in Eq. (126), is obtained:
where the integral in Eq. (128), can be interpreted as a quantum similarity measure , that is:
The unknown coefficient set in Eq. (128): \( |\omega\rangle =\{\omega_I|I=1,M\} \), which can be collected into an M-dimensional column (or row) vector, will represent the operator W in terms of the known density function basis set B. This situation, clearly represented by Eq. (128), still has a set of undetermined parameters, associated now to the vector \( { |\omega\rangle } \) components, instead of the operator W.
Equation (128) can be used to obtain the vector \( { |\omega\rangle } \). As is usually the case in classical QSPR, it is only necessary to know, a quantum object tag set, associated to some molecular probe set P of cardinality N, \( P=\big\{\rho_J^P \wedge p_J|J=1,N\big\} \), where, as previously commented, every quantum object structure in P has also necessarily to be attached to a known value of the involved observable: \( |p\rangle = \{p_J|J=1,N\} \), which can be also collected in form of a N-dimensional column (or row) vector. Then, Eq. (128) can be rewritten for every element in P, employing the known property values instead of the expectation observable values, that is:
in this way the following set of quantum similarity measures is generated:
which in turn can be considered, after an appropriate rearrangement, as elements of a \( { (M\times N) } \) similarity matrix, involving the basis and probe quantum object molecular sets respectively: \( { \mathbf{A}=\big\{a_{IJ}^{BP}\big\} } \).
With this matrix definition in mind, then Eq. (130) can be rewritten as a linear system in matrix form, connecting the already defined vectors in row space form:
Such a linear system can be associated to the most common dual problem in column vector space, just defining the transpose of the similarity matrix, using the usual definition:
and in this manner, the fundamental QQSPR equation is set up, writing a column equivalent dual expression of the former row Eq. (132):
As in classical QSPR, the solutions of Eq. (134) may provide the knowledge of the coefficient vector \( { |\omega\rangle } \). However, it must again be stressed that Eq. (134) differs from the classical QSPR setup in the sense that such an equation can be deduced from the quantum mechanical statistical structure, associated to expectation value calculations. In this way, the causal connection between molecular structure and molecular properties can be deduced from employing quantum mechanical theoretical fundaments, via the ideas of quantum similarity. The interest of such a relationship lies in the fact that fundamental QQSPR equations can be extended to any quantum object structure and properties. So, obviously, these relationships can be applied to molecular systems as well, provided they can be described as quantum objects, making QQSPR universal in the sense that it can be applied, under the same conditions, to any sub-microscopic quantum object set.
Non-Linear QQSPR Equations .
In a second remark step, which appears to be sufficiently important as to merit a separate section treatment, the approximate operator linear description (127) may be extended with non-linear terms, which can be easily provided by the nature of the involved quantum object density function tags, which can be founded in turn on the theoretical development of extended wave functions.
In this case, Eq. (127), can be written in a more structured manner as a truncated Taylor series, where only the first two terms are kept for simplicity:
however, with the potential prospect to add terms up to any order. Equation (135) can be perhaps also considered a simplification of a series involving density functions of growing orders, that is:
The second-order coefficient set \( { \{\omega_{PQ}\} } \) in Eq. (135), can be also substituted as well, in order to retain a minimal number of unknowns, by products of first-order coefficients, in the following way:
Then, just if this is the case, Eq. (130), transforms into a more computationally convenient form:
Triple Density Quantum Similarity Integrals.
The integrals included in the second-order terms of Eq. (138) are triple density similarity measures , which can have the form chosen, among many other possibilities, in the following way:
Moreover, the usual computational form of the triple density measures can be the one, where the operator becomes unit and all the integrand density functions bear the same variable, so the integral in Eq. (139) acquires a simpler structure, like the triple density overlap integral form:
while, first-order similarity measures (129) become, under an equivalent simplification, overlap-like integrals:
Equations (140) and (141), could be obtained defining the respective weighting operators in terms of an integral operator, involving as many products of Dirac's delta functions as density functions appear into the integrand. For instance, in Eq. (139), the operator \( { \Theta(\mathbf{r}_1;\mathbf{r}_2;\mathbf{r}_3) } \) can be substituted inside the integral in the following manner:
It is, then, straightforward to use the same technique to obtain equations possessing a higher number of density function terms, and so it is easily seen how to take into account and to handle them in the same manner, adding higher order terms within non-linear fundamental QQSPR equations of type (138).
Hansch-Type QQSPR Quadratic Models.
In the same manner as above in the linear case, the fundamental quadratic QQSPR Eq. (138) can be simplified, so only the diagonal terms of the initial equation remain. First using just a probe set, taking \( { B=P } \) and then supposing that the remnant equation summation terms are constant under the study of some quantum objects, possessing a great deal of homogeneity. In this case one can write:
which constitutes a quadratic extension of the linear Hansch-type relationships .
Quadratic Fundamental QQSPR Equation in Matrix Form.
Having set up in the way outlined above the formal structure of the fundamental QQSPR equations, we now need to discuss its matrix implementation, which is an obligatory step when seeking computational algorithms in practical cases. Two possible equivalent modes will be discussed in this section: the first one corresponds to classical matrix product formalism, while a second part will present an equivalent form just employing inward matrix products. The reason for this second formal presentation is the easiness of setting a general framework up to any approximation order.
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a)
Classical Form
Equation (138) can be easily written in matrix form. For this purpose it is only necessary to define, besides the column vector of the first-order coefficients:
$$ |\omega\rangle = \{\omega_I|I=1,M\}\:; $$(144)also, for every quantum object within the probe set, the first-order M-dimensional similarity matrix columns:
$$ J=1,N\colon \big|\mathbf{z}_{IJ}^{(1)}\big\rangle = \big\{z_{IJ}^{(1)} = \big\langle\rho_I^B|\Theta|\rho_J^P\big\rangle|I=1,M\big\}\:, $$(145)as well as the second-order \( { (M\times M) } \)-dimensional similarity matrices:
$$ J=1,N\colon\\ \big|\mathbf{Z}_J^{(\mathbf{2})}\big\rangle = \big\{z_{J;PQ}^{(2)} = \big\langle\rho_J^P|\Theta|\rho_P^B\rho_Q^B\big\rangle|P,Q=1,M\big\}\:, $$(146)shall be constructed.
Taking the above-defined similarity matrices into account, Eq. (138) can be written as:
$$ J=1,N\colon p_J \simeq \big\langle\mathbf{z}_J^{(\mathbf{1})}|\omega\big\rangle + \big\langle\omega|\mathbf{Z}_J^{(\mathbf{2})}|\omega\big\rangle + O(3)\:, $$(147)so, collecting the property observable values into a column vector, as already discussed and then, reordering first- and second-order matrix components in the following way:
$$ \mathbf{Z}^{(1)} = \big\{\big|\mathbf{z}_J^{(1)}\big\rangle|J=1,N\big\}\:, $$(148)and
$$ \mathbf{Z}^{(\mathbf{2})} = \big\{\mathbf{Z}_J^{(\mathbf{2})}|J=1,N\big\}\:, $$(149)then the second-order fundamental QQSPR equation becomes a quadratic system of equations in matrix form:
$$ |p\rangle \simeq (\mathbf{Z}^{(1)} + [\langle\omega|\mathbf{Z}^{(2)}])|\omega\rangle +O(3)\:. $$(150) -
b)
Inward Matrix Product Form as a Generalization Device
Alternatively, there is the possibility to express the equations of the previous description by means of inward matrix products . The first-order term in Eq. (147) can be expressed within inward product formalism at once, as it is a simple scalar product between the involved vectors, so:
$$ J=1,N\colon \big\langle\mathbf{z}_J^{(\mathbf{1})}|\omega\big\rangle \equiv \big\langle\big|\mathbf{z}_J^{(\mathbf{1})}\big\rangle\ast|\omega\rangle\big\rangle\:, $$(151)while the second-order term may be expressed in inward product form with the aid of the coefficient vector tensor product, forming a square \( { (N\times N) } \) matrix:
$$ \mathbf{W} = |\omega\rangle \otimes |\omega\rangle \equiv \{w_{IJ} =\omega_I \omega_J |\forall I,J=1,N\}\:, $$(152)so, one can write then the quadratic term of Eq. (150) as an inward matrix product too:
$$ J=1,N\colon\\ \big\langle\omega|\mathbf{Z}_J^{(\mathbf{2})}|\omega\big\rangle \equiv \big\langle \mathbf{Z}_J^{(\mathbf{2})}\ast \mathbf{W} \big\rangle = \big\langle\mathbf{Z}_J^{(\mathbf{2})}\ast(|\omega\rangle \otimes|\omega\rangle)\big\rangle\:, $$(153)and consequently Eq. (147), can be rewritten as:
$$ J=1,N\colon\\ p_J \simeq \big\langle\big|\mathbf{z}_J^{(\mathbf{1})}\big\rangle\ast|\omega\rangle\big\rangle + \big\langle\mathbf{Z}_J^{(\mathbf{2})} \ast (|\omega\rangle \otimes |\omega\rangle)\big\rangle +O(3)\:. $$(154)
Inward Matrix Product Formalism of Fundamental QQSPR Equation nth Order Terms.
Both, classical and inward product, formalisms are equivalent; however, the inward product Eq. (154), permits one to easily imagine any sequence of corrections into the fundamental QQSPR equation, up to any arbitrarily chosen nth order term, just writing:
where the leading equation terms are \( { \big\{\mathbf{Z}_J^{(R)}|J=1,N\big\} } \) the Rth order similarity matrices, which can be constructed as:
with the index set: \( { S(\mathbf{i})=\{S_1 ,S_2 ,\ldots, S_R\} } \) formed by any of the M R combinations with repetition of R elements chosen within the M integers and, finally, the Rth order tensor products of the coefficient vector are noted as: \( { \mathop{\otimes}_{S=1}^R|\omega\rangle } \).
Stochastic Transformations.
We will now discuss a third remark step, dealing with the stochastic transformation of similarity matrices, because it also merits a separate section. Recently, several studies have dealt with stochastic transformations of the fundamental QQSPR equation in linear symmetric form, that is: using \( { B=P } \).
At the light of the previous manipulation presented in this study, the stochastic structure transformation of the fundamental QQSPR equation has to be performed, at any operator-equation approximation level, using the possibility to compute the sum of the elements of the Rth order similarity matrices as have been previously defined in Eq. (156), that is:
where a nested summation symbol \( { \sum(\mathbf{i}) } \) has been employed in order to indicate the nested sums over the R indices, represented by the index sets: \( S(\mathbf{i}) = \{S_1 ,S_2 ,\ldots, S_R\} \). Using the sum of the similarity matrix elements (157), then the elements of the new matrices scaled by this sum become scaled in turn as follows:
and the new Rth order stochastic similarity matrices behave as a discrete probability distribution, as: \( \forall S(\mathbf{i})\colon z_{J;S(\mathbf{i})}^{(R)} \in \mathbf{R}^+\to s_{J;S(\mathbf{i})}^{(R)} \in \mathbf{R}^+ \) and besides:
Both properties can be cast into a unique convex condition symbol:
So, in this way, the stochastic matrix set: \( \mathbf{S}=\big\{\mathbf{S}_J^{(R)}|R=1,n\big\} \) can be considered, up to nth order, as a set of M R-dimensional unit shell elements, belonging to some vector semispace with the same dimensions. In these circumstances one can consider the fundamental QQSPR Eq. (155) as to be written:
where everything is the same as in the former Eq. (155), except for the similarity matrix set, which has been substituted by the stochastic matrices (158).
The coefficient vector has been left unchanged, but evidently its character could be no longer the same as in Eq. (155). However, the nature of the coefficient vector can be more precise in this case of the fundamental QQSPR stochastic Eqs. (161). This is due to the characteristic convex condition properties, which possess the semispace unit shell elements obtained transforming the similarity matrices.
In fact, the stochastic similarity matrix set: \( \big\{\mathbf{S}_J^{(R)}|R=1,n\big\} \), so naturally obtained from the original similarity matrix set, can be interpreted as a sequential discrete representation of the continuous normalized density function, associated to the involved Jth quantum object. Then, from the quantum mechanical point of view, the whole stochastic matrix set can be viewed as a discrete quantum object tag collection. Thus, in this case, the tensor products of the coefficient vector can be easily considered as arrays of convex sets, that is:
because, whenever the generating coefficient vector is a convex vector, that is, fulfilling the convex conditions:
then, any tensor product of the convex vector \( { |\omega\rangle } \) fulfils: \( { K\big(\mathop{\otimes}_{S=1}^R|\omega\rangle\big) } \). Indeed, if convex conditions (163) hold, then it is easy to see that convex conditions are present within any arbitrary order tensor product of the coefficient vector, as shown in the following deduction:
Variational QQSPR .
So far the fundamental QQSPR equation has been solved using the usual strategy associated to classical QSPR. Equations (134), (150) or (161) as in classical terms, can be solved for the coefficient vector \( { |\omega\rangle } \). As has been previously commented this is done, by substituting in the expectation value expression (138) the vector \( { |\pi\rangle } \) by an experimental property vector \( { |p\rangle } \), associated to the probe quantum object set P. The result will be obtained in the same way as in classical QSPR, but using the quantum similarity matrices as molecular descriptors. However, it can be proven that the fundamental QQSPR equation can be solved within the usual quantum variational procedures.
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a)
Similarity Matrix Unrestricted Variational Treatment
For such a purpose it is sufficient to rewrite the second-order expectation value Eq. (138) as:
$$ \forall J=1,N\colon\\ \langle\pi_J\rangle \simeq \sum\limits_{P=1}^M \omega_P z_{J;P}^{(1)} +\sum\limits_{P=1}^M \sum\limits_{Q\geq P}^M \omega_P \omega_Q z_{J;PQ}^{(2)} + O(3) $$(165)then, considering every quantum object expectation value as a variational function of the parameters within the coefficient vector \( { |\omega\rangle } \), the resulting expression can be varied, taking into account that the density functions, supposedly obtained by quantum mechanical procedures, no longer need variation. In this way, every Jth quantum object will have to possess a specific coefficient vector \( { |\omega\rangle } \), which can be thus named as \( { |\omega_J\rangle } \). That is:
$$ \forall J=1,N\colon \delta\langle\pi_J\rangle \simeq \sum\limits_{P=1}^M \delta\omega_P z_{J;P}^{(1)}\\ +2\sum\limits_{P=1}^M \sum\limits_{Q\geq P}^M \omega_P \omega_Q z_{J;PQ}^{(2)} +O(3)\:, $$(166)then, using the variation condition for the Jth quantum object:
$$ \delta\langle\pi_J\rangle =0\:, $$(167)is obtained:
$$ \forall J=1,N\colon 0\simeq \sum\limits_{P=1}^M \delta\omega_P z_{J;P}^{(1)}\\ +2\sum\limits_{P=1}^M \sum\limits_{Q\geq P}^M \delta\omega_P \omega_Q z_{J;PQ}^{(2)} +O(3)\:, $$(168)which can be rewritten as:
$$ \forall J=1,N\wedge P=1,M\colon\\ 0\simeq z_{J;P}^{(1)} + 2\sum\limits_{Q=1}^M \omega_Q z_{J;PQ}^{(2)} +O(3)\:. $$(169)This last equation can be expressed in matrix form, using the appropriate similarity matrices as previously defined in Eqs. (145) and (146):
$$ \forall J=1,N\colon \mathbf{z}_J^{(1)} +2\mathbf{Z}_J^{(2)} |\omega_J\rangle =0\:, $$(170)thus, the specific coefficients for each quantum object may be computed as:
$$ \forall J=1,N\colon |\omega_J\rangle = -\frac{1}{2}\big[\mathbf{Z}_J^{(2)}\big]^{-1}\mathbf{z}_J^{(1)}\:. $$(171)This is the same as associating a particular operator W to each quantum object, and such a result is not too surprising a feature, as the operator W can be easily supposed to vary from one quantum object to another, in the same way as Hamilton operators do. The variational expectation value for the Jth object could be obtained in this case as:
$$ \langle\pi_J\rangle \simeq \big\langle\omega_J|\mathbf{z}_J^{(1)}\big\rangle +\big\langle \omega_J|\mathbf{Z}_J^{(2)}|\omega_J\big\rangle +O(3)\:. $$(172)Using Eq. (171) into Eq. (172), the following expectation value final optimal form will result:
$$ \langle\pi_J\rangle \simeq -\frac{1}{4}\big\langle\mathbf{z}_J^{(1)}\big|\big[\mathbf{Z}_J^{(2)}\big]^{-1}\big|\mathbf{z}_J^{(1)}\big\rangle +O(3)\:. $$(173) -
b)
Expectation Versus Experimental Values
Then, the set of stationary expectation values \( { |\pi\rangle } \) can be compared with the experimental value vector \( { |p\rangle } \), in such a way as to have:
$$ |p\rangle =a+b|\pi\rangle\:, $$(174)\( { \{a,b\} } \) being some origin and scale parameters, respectively. They can be obtained by the usual well-known regression techniques.
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c)
Algorithm for Unrestricted Variational QQSPR
Once the set of coefficients \( { \{a,b\} } \) is obtained by using Eq. (174) for a given probe quantum object set, the property expectation value \( { \langle\pi_K\rangle } \) of any new quantum object K, say, with known density function \( { \rho_K } \), can be employed to estimate the experimental value ? K of the quantum object studied property, by using the following steps:
-
1.
Compute: \( { \big\{\mathbf{z}_K^{(1)};\mathbf{Z}_K^{(2)}\big\} } \) using the basis set B.
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2.
Evaluate: \( { \langle\pi_K\rangle \simeq -\frac{1}{4}\big\langle\mathbf{z}_K^{(1)} \big| \big[\mathbf{Z}_K^{(2)}\big]^{-1} \big|\mathbf{z}_K^{(1)}\big\rangle +O(3) } \)
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3.
Obtain the estimated property: \( { p_K =a+b\langle\pi_K\rangle } \).
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1.
Stochastic Similarity Matrices Restricted Variational Treatment.
Of course, all that has been said up to now in this section remains valid for stochastic similarity matrices: \( \big\{\mathbf{s}_K^{(1)};\mathbf{S}_K^{(2)}\big\} \), they just have to be used instead of the similarity matrix pair: \( { \big\{\mathbf{z}_K^{(1)};\mathbf{Z}_K^{(2)}\big\} } \) in the above algorithm. However, the stochastic case may be interesting if the coefficient set \( { |\omega\rangle } \) can be obtained obeying convex conditions as a restriction, so that the previous unrestricted variation algorithm may no longer be applicable.
Expectation Value Jacobi Rotations Variational Form.
To obtain the desired restricted variation over the coefficient vector involved in expectation value expressions, a similar procedure as the one employed in developing the ASA technique [75,76,77,78,79,80,81,82,83] could be easily set up to perform the variational computation over Eq. (165), but taking into account the additional restriction of obtaining a convex vector, as a result of the optimization process.
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a)
Preliminary Considerations
When this option as discussed above is chosen, it is only necessary to express the operator W variational coefficients with the aid of a new free normalized auxiliary vector; in order to ensure the convex conditions hold throughout the entire optimization process, that is:
$$ \begin{aligned} |\omega\rangle &= |x\rangle\ast|x\rangle \wedge\langle x|x\rangle = 1 \to \langle|\omega\rangle\rangle\\ &= \sum\limits_I \omega_I = \sum\limits_I x_I^2 =1\wedge \forall I\colon \omega_I = x_I^2 \in \mathbf{R}^+\:. \end{aligned} $$(175)After this consideration, it is only necessary to obtain the variation of Eq. (165), by applying norm conserving, orthogonal elementary Jacobi rotations [148] into the auxiliary vector \( { |x\rangle } \) element pairs, in order to arrive at an expression, depending on the elementary Jacobi rotation angle, which could be easily optimized later on.
An interesting point at this stage is to realize that such a restricted variational procedure can be applied to higher order equations, with orders larger than the ones studied up to now. This is due to the fact that Jacobi rotations over the auxiliary vector just change a couple of the coefficient auxiliary vector elements each time an elementary Jacobi rotation is performed, and the same occurs with the coefficient vector. This knowledge of the coefficient vector variation can be easily brought into the tensor products and worked out up to any tensor order.
The rest becomes a procedure with somehow a growing technical computational complexity, but defined within a well-structured theoretical background algorithm.
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b)
Elementary Jacobi Rotations Algorithm Scheme
Elementary Jacobi rotations need the cosine, c, and the sine, s, of a rotation angle. These involved trigonometric functions fulfil the usual convex relationship: \( { c^2+s^2=1 } \). When acting over a vector, the Jacobi rotations will change two vector components, the Kth and Lth, say, leaving the remaining components as they are:
$$ \begin{aligned}[b] |x\rangle &= \begin{pmatrix} \cdots \\ x_K \\ \cdots \\ x_L \\ \cdots\end{pmatrix} \to \begin{pmatrix} \cdots \\ cx_K -sx_L \\ \cdots \\ sx_K +cx_L \\ \cdots\end{pmatrix}\\ \Rightarrow |w\rangle &=\begin{pmatrix} \cdots \\ x_K^2 \\ \cdots \\ x_L^2 \\ \cdots\end{pmatrix} \to \begin{pmatrix} \cdots \\ (cx_K -sx_L)^2 \\ \cdots \\ (sx_K +cx_L)^2 \\ \cdots\end{pmatrix}\:. \end{aligned} $$(176)It is easy to obtain the variation in the coefficient vector due to an elementary Jacobi rotation as:
$$ |\delta\omega\rangle = v_{KL} \begin{pmatrix} \cdots \\ -1 \\ \cdots \\ +1 \\ \cdots\end{pmatrix} = v_{KL} (|e_L\rangle - |e_K\rangle)\:, $$(177)where \( { \{|e_K\rangle,|e_L\rangle\} } \) are the corresponding canonical basis set vectors. The scalar coefficient v KL possesses the form:
$$ v_{KL} = s^2\big(x_K^2 -x_L^2\big) + 2 csx_K x_L\:. $$(178)Then, employing this result in the equivalent expression of Eq. (147), but written in expectation value matrix form, the following can be deduced:
$$ \langle\delta\pi\rangle = \langle\delta\omega|(|\mathbf{z}^{(1)}\rangle + 2\mathbf{Z}^{(2)}|\omega\rangle) + \langle\delta\omega|\mathbf{Z}^{(2)}|\delta\omega\rangle\:, $$(179)where the quantum object subindex has been taken out to simplify the notation. Then, upon substituting the coefficient vector variation:
$$ \langle\delta\pi\rangle = v_{KL} \bigg[\big(z_L^{(1)} -z_K^{(1)}\big) + 2\sum\limits_I \omega_I \big(Z_{IL}^{(2)} -Z_{IK}^{(2)}\big)\bigg]\\ + v_{KL}^2 \big(Z_{KK}^{(2)} + Z_{LL}^{(2)} -2Z_{KL}^{(2)}\big) $$(180)which, upon equalization to zero and terms rearrangement, can be expressed as a second-order equation on the elementary Jacobi rotation sine and cosine:
$$ As^2+Bsc+\beta =0\:, $$(181)with the coefficients A and B defined as:
$$ \begin{aligned} A &= \alpha(\omega_K -\omega_L) \\ B &= 2\alpha x_K x_L \end{aligned} $$(182)and, besides, the parameters are constructed by the elements of the similarity matrices in the following way:
$$ \begin{aligned} \alpha &= Z_{KK}^{(2)} + Z_{LL}^{(2)} - 2Z_{KL}^{(2)} \\ \beta &= \big(z_L^{(1)} -z_K^{(1)}\big) + 2\sum\limits_I \omega_I \big(Z_{IL}^{(2)} -Z_{IK}^{(2)}\big)\:. \end{aligned} $$(183)
Higher Order Stochastic Expectation Value Variational Treatment.
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a)
General Comments
Whenever Eq. (161) is studied, after being conveniently modified for the expectation values form,
$$ \forall J=1,N\colon\langle\pi_J\rangle = \sum\limits_{R=1}^n \big\langle\mathbf{S}_J^{(R)}\ast\mathbf{W}^{(R)}\big\rangle +O(n+1) $$(184)the obvious fact appears that the variation will affect just the Rth order tensor products \( { \mathbf{W}^{(R)} } \) of the coefficient vector. So it can be written, dropping the quantum object subindex J just for convenience, as before:
$$ \langle\delta\pi\rangle = \sum\limits_{R=1}^n \langle\mathbf{S}^{(R)}\ast\delta\mathbf{W}^{(R)}\rangle +O(n+1)\:, $$(185)so the relevant variation will be associated to the terms \( { \delta\mathbf{W}^{(R)} } \), which can be easily written, using a tensor notation as:
$$ \begin{aligned} \delta\mathbf{W}^{(R)} &= \delta \Big(\mathop{\otimes}\limits_{S=1}^R |\omega\rangle\Big)\\ &= \sum\limits_{S=1}^R \begin{pmatrix} R \\ S\end{pmatrix} \Big[\Big(\mathop{\otimes}\limits_{P=1}^{R-S} |\omega\rangle\Big) \otimes \Big(\mathop{\otimes} \limits_{Q=1}^S |\delta\omega\rangle \Big)\Big]\:, \end{aligned} $$(186)but being the definition of the coefficient vector variation, upon Jacobi rotations, well known from Eq. (177), it can be written:
$$ \delta\mathbf{W}^{(R)} = \sum\limits_{S=1}^R \begin{pmatrix} R \\ S\end{pmatrix} (v_{KL})^S \\ \cdot\Big[\Big(\mathop{\otimes}\limits_{P=1}^{R-S}|\omega\rangle\Big)\otimes \Big(\mathop{\otimes}\limits_{Q=1}^S[(|e_L\rangle -|e_K\rangle)]\Big)\Big]\:. $$(187)So in this way, the restricted variation of the expectation value QQSPR equations, using elementary Jacobi rotations, is clearly defined up to any order.
-
b)
A Computational Detail Concerning Tensor Products of the Difference of Two Canonical Vectors
The tensor product of the difference between the pair of canonical basis set vectors:
$$ |e_L\rangle -|e_K\rangle = \begin{pmatrix} \cdots \\ -1 \\ \cdots \\ +1 \\ \cdots\end{pmatrix} \equiv |L\rangle -|K\rangle \equiv |L-K\rangle\:, $$(188)which appears in Eq. (187), may be expressed in terms of a nested summation symbol. For example, up to second order the sum of the four tensor terms is readily written as:
$$ |L-K\rangle \otimes |L-K\rangle\\ = |L\otimes L\rangle - | L\otimes K\rangle - |K\otimes L\rangle + |K\otimes K\rangle $$(189)with the obvious meaning for the involved tensors:
$$ |L\otimes L\rangle = |e_L\rangle \otimes |e_L\rangle = \mathbf{E}_{LL} = \{e_LL;PQ = \delta_{LP} \delta_{LQ}\} $$(190)and so on.
In general, up to Sth order:
$$ \mathop{\otimes}\limits_{Q=1}^S |L-K\rangle = \sum(\mathbf{i}) \sigma(Q(\mathbf{i}))|Q(\mathbf{i})\rangle\:, $$(191)where \( { Q(\mathbf{i})=\{Q_1 \otimes Q_2 \ldots \otimes Q_S\} } \) is any of the possible \( { 2^n } \) combinations with repetition of the indices K and L, the symbol \( { |Q(\mathbf{i})\rangle } \) meaning a tensor product of the initial canonical basis set vectors with such an index repetition. That is: an object equivalent to a canonical hypermatrix, whose elements are all zero, except the one with indices associated to those entering the set. Also \( { \sigma(Q(\mathbf{i})) } \) corresponds to the sign, associated to the fact that the index K appears in \( { Q(\mathbf{i}) } \) an even, \( { \sigma(Q(\mathbf{i}))=+1 } \), or odd, \( { \sigma(Q(\mathbf{i}))=-1 } \), number of times.
QQSPR Operators, Quantum Similarity Measures and the Fundamental QQSPR Equation
The correspondence principle in quantum theory furnishes the rules to construct Hermitian operators, whose expectation values can be associated withthe experimental outcomes of submicroscopic system observables. However, as has been previously commented, for some observables of complex submicroscopicsystems, like some biological activities of pharmaceutical interest, the correspondence principle cannot be applied. The construction of the QQSPRoperators and the attached fundamental QQSPR equation provide the possibility to attach an approximate quantum mechanical operator to estimate expectationvalues for these cases.
The QQSPR Operator.
The fundamental QQSPR equation arises when from the known quantum objects, belonging to some quantum object set; one realizes that their density function tags: \( { \{\rho_I(\mathbf{r})\} } \) can be used to construct a QQSPR operator in the form:
in the Eq. (192) above, w 0 is an arbitrary constant; \( { \{\Theta_\omega(\mathbf{R})|\omega =0,1,2,\ldots\} } \) is a known positive definite operator set, acting as a weight for each term development; finally, \( { \{w_I\} } \) a set of unknown parameters which shall be determined through the fundamental QQSPR equation as will be explained below.
Thus, the structure of a QQSPR operator like the one defined in Eq. (192) has to be seen as the first step of an algorithm permitting the construction of approximate quantum mechanical operators, associated in turn to some observables of complex submicroscopic systems, whose nature do not permit the application of the correspondence principle to construct Hermitian operators for the evaluation of observable values.
The Expectation Values of the QQSPR Operator.
In order to determine the parameter set \( { \{w_I\} } \), defining in this way the QQSPR operator as written in Eq. (192), it is just necessary to compute the set of expectation values over the elements of a quantum object set which belong to the core set C, constituted by the core molecules or C-m. Besides a well-defined structure and a known density function, as members of a quantum object set, the C-m are supposed to possess an element of a known property set \( { P=\{p_K\} } \), attached to each one.
In this way one can express every known property of the C-m elements as the expectation value of some QQSPR operator:
Zero-th Order Term.
When describing the expectation values of the C-m as computed in Eq. (193), one can consider first the Zero-th order term:
as being a constant for each C-m, which can be used as an origin shift of the C-m property tags, thus the Zero-th order term: \( { w_0 \Theta_0(\mathbf{r}) } \) appearing in the above operator definition acts as a gauge. Choosing the Zero-th order operator as the unit, this term becomes proportional to the number of electrons of the C-m considered:
In case shape functions, defined as:
are employed in the QQSPR operator definition (192) and in the expectation value expression (193), then the Zero-th order contribution to the expectation values \( { \theta_K[I] } \) is a constant for all C-m.
The Zero-th order term can be omitted if it is no longer necessary to shift the property values of the C-m.
First- and Second-Order Expectation Value Terms.
The first-order term of the expectation value Eq. (193) contains quantum similarity measure integrals among pairs of density function tags of the C-m, which have been defined a long time ago as:
and in the second-order term the triple density quantum similarity measures appear, defined as well as:
Fundamental QQSPR Equation Setup.
The expectation values of the QQSPR operator, as described in Eq. (193), can be collected in a column vector providing the fundamental QQSPR equation:
where in Eq. (194) the following compact symbols have been used: \( { |\mathbf{p}\rangle = \{p_K\} } \) is the C-m properties vector, \( { |\theta\rangle = \{\theta_K\} } \) is the completely determined gauge shift vector, \( { \{\mathbf{Z}_\omega|\omega =1,2,\ldots\} } \) is a matrix set containing the quantum similarity measures, for instance: \( \mathbf{Z}_1 = \{z_{IK}\} ; \mathbf{Z}_2 = \{z_{IJK}\};\ldots \), and \( { |\mathbf{w}\rangle = \{w_I\} } \) is a column vector bearing the unknown coefficients, which define explicitly the QQSPR operator.
The easiest way to obtain the unknown coefficients \( { |\mathbf{w}\rangle = \{w_I\} } \) is obviously the linear equation contained in the fundamental QQSPR Eq. (194); that is, they can be evaluated by solving:
However, Eq. (195) has no predictive power whatsoever. This is so because the first-order similarity matrix \( { \mathbf{Z}_1 } \) has to be chosen positive definite by construction, therefore the coefficient vector has a unique determined form.
By predictive power is meant here the possibility to compute the value of the property, which precisely defines the C-m set, for an also known quantum object, which as such possesses well-defined structure and density function, but belongs to the Unknown property molecular set: U, whose elements are made by unknown property molecules or quantum objects, the U-m.
In the last years, since the description of quantum similarity measures, the predictive power of the information contained in the similarity matrices set has been manipulated in the classical QSPR way. For example, using similarity matrices principal components, and finding with them a QSAR model, usually multilinear. This multilinear model can be employed, afterwards, to estimate U-m properties. This amounts to the same as considering the similarity matrices as a source of molecular parameters to construct empirical QSPR.
However, there is a possible way to use the system (195) for predicting properties of U-m without further considerations than the involved algebraic procedures. The possible QQSAPR prediction algorithms will be developed in a separate section.
Evaluation of Unknown Molecular Properties as Expectation Values
In general, one can choose any molecular structure U, possessing an unknown value of the property needed to build up the core set triads. Thereafter, one can call such a QO the U-molecule or U-m, for sake of simplicity. The U-molecule can supposedly be associated to a corresponding density function:\( { \rho _U } \) too. Hence, the U-m can be certainly considered as a QO. On the other hand, one must keep in mind that, by construction of the QSPR problem, the property lacking in the information about U has to be already known for all elements of the core set. One can easily express an approximate value of the U-m unknown property through the simplified Minkowski norm:
provided that the set of coefficients \( { \{x_I\} } \) is well-defined.
The Minkowski norm in Eq. (196) can be computed in more sophisticated ways, using a known positive definite operator, W say, as a weight in the expectation value definition: \( { \langle\Omega W[\rho_U]\rangle } \), producing weighted quantum similarity measures of type: \( { \langle\rho(\mathbf{i})W[\rho_U]\rangle } \) in the right part of the expression (196). In order to simplify the formalism, here the convention: \( { W=I } \), has been adopted.
Within the QQSPR problem settings, the set of coefficients: \( { \{x_I\} } \), in Eq. (196), which can be ordered as a column vector: \( { |\mathbf{x}\rangle = \{x_I\} } \), is unknown beforehand, but can be already computed from the first-order approach using the core set known property values, as will be discussed below.
Quantum Similarity Matrices in the Construction of First-Order QSPR Operators and the Definition of Discrete QOS.
The first-order approach of the QSPR operator for the core set known molecular property tag set: \( { \Pi =\{\pi_I\} } \) generates the following equation collection:
The set of integrals:
appearing in Eqs. (197) can be ordered into a \( { (n\times n) } \) symmetric array, constructing in this way the quantum similarity matrix : \( { \mathbf{Z} = \{z_{IJ}\} } \) (QSM). In turn, the ordered set of shifted properties: \( { \{p_I\} } \) can form a \( { (n\times 1) } \) column vector: \( { |\mathbf{p}\rangle = \{p_I\} } \). Therefore, the equation set (197) is simply a linear system, which will be discussed next, in order to describe its possible use for evaluating U-m unknown molecular properties.
Empirical QSPR.
In the empirical QSPR problems, the equivalent matrix to the QSM of the QQSPR framework, as described in Sect. “Quantum Similarity”, can be obtained in the following manner. Suppose that every molecular structure of M possesses an arbitrarily chosen empirical descriptor vector, in that way:
then the descriptor set D acts as a tag set to construct an empirical discrete tagged set:
such that:
The discrete tag set D of molecular descriptors can be considered, in turn, as a linearly independent subset of cardinality n belonging to some real m-dimensional column vector space, that is: \( { \text{D} \subset V_m (\mathbf{R}) } \). The linear independence of the set D is strictly necessary to construct a matrix comparable in properties to QSM, and in this way, each molecule becomes independently described from the rest. With this information in mind it is easy to construct, \( { \mathbf{S}_{\text{D}} } \), a symmetric \( { (n\times n) } \) matrix bearing analogous characteristics as the QSM:
In fact, constructing the \( { (m\times n) } \) matrix: \( { \mathbf{D} = \{|\mathbf{d}_I\rangle\} } \), whose columns are the elements of the empirical descriptor set D, then the matrix \( { \mathbf{S}_{\text{D}} } \) can also be defined as the product:
where \( { \mathbf{D}^\mathrm{T}=\{\langle\mathbf{d}_I|\} } \) is the \( { (n\times m) } \) transpose of matrix D, whose rows are the descriptor vectors ordered in such a way. It is easy to see that matrix \( { \mathbf{S}_{\text{D}} } \), defined in this manner, is coincident with the Gramian matrix of the tag set D. In order to comply with the same standard properties as the QSM the matrix \( { \mathbf{S}_{\text{D}} } \) has to fulfil: \( { \operatorname{Det}|\mathbf{S}_{\text{D}}| > 0 } \). If this is the case, a discrete empirical object set \( { \text{Q}_{\text{S}} } \) can be defined as:
in close resemblance to the discrete quantum object set \( { \text{Q}_{\text{Z}} }\) described in Eq. (42).
Finally, one shall comment now that, as a consequence of this definition of the set \( { \text{Q}_{\text{S}} } \), the presentation and discussion about the following procedures, which will be studied in this paper for QSM, can also be applied to the Gramian matrices, associated to empirical descriptor tag sets and thus to the classical QSPR problem.
However, the different background between quantum and empirical points of view induces the necessary emergence of the following considerations. The QQSPR equations are deductible from the usual quantum theoretical considerations; within the same context, they can be easily generalized to contain higher approximation orders. Therefore, the QQSPR equations of any order can certainly possess in general some causal background; while, except for very particular cases, empirical QSPR equations remain arbitrarily constructed and without a clear causal fundament.
First-Order Fundamental QQSPR (FQQSPR) Equation
The analysis of the QQSPR problem can start with the first order or linear fundamental QQSPR equation, involving the core set, formed with the molecules of the associated DQOS, which are also linked with known values of some property, according to the considerations noted above.
One can write Eq. (197) in a compact matrix form:
Where the matrix Z is the already described symmetric QSM, \( { |\mathbf{p}\rangle } \) is the known core set property vector and \( { |\mathbf{x}\rangle } \) is a \( { (n\times 1) } \) vector, whose coefficients have to be evaluated.
The predictive power of such an equation is a priori null, because being the QSM: Z, by construction non-singular (otherwise two density functions will be exactly the same), then there always can be computed a QSM inverse: \( { \mathbf{Z}^{-\mathbf{1}} } \), obeying the usual relationships: \( { \mathbf{Z}^{-1}\mathbf{Z}=\mathbf{ZZ}^{-1}= \mathbf{I} } \), in such a way that the trivial result, defining the unknown coefficient vector:
will be always obtained within a core set scenario. Furthermore, one can retrieve the exact value of the property for any molecule of the core set QOS choosing the scalar products:
The QSM for diverse core sets has been used in a quite large set of prediction studies [26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46], in every case employing up to date statistical tools, the usual procedures currently available in classical QSPR studies, see for example reference [171]. The use of the first-order fundamental QQSPR equation to construct algorithms, which can be utilized as predictive tools, has been previously attempted [59], but it has not been continued in practice. In the present study, the reader can find in the following sections new theoretical developments of the prediction ability of the fundamental QQSPR equation . However, a reminder of some simple linear algebra relating to the FQQSPR equation is needed first in order to understand the following arguments; therefore it will be described in the forthcoming final section.
Future Trends
Procedure for Adding One Molecular Structure to a Known Core Set
The Partition of the FQQSPR Linear Equation.
The general setup described until now, amounts to the same as virtually considering the U-m as forming part of the core set, but with a parametrized value of the unknown property. One can refer to this extension of the core set as the parametrized core set .
The following coefficient vector partitioned expression can be easily written in terms of the inverse partitioned QSM matrix elements:
However, in order to obtain equivalent expressions possessing less entanglement with the elements of the inverse matrix, the most convenient way is to restart the procedure writing explicitly:
From the augmented linear equation first component structure, one can obtain:
taking into account that the QSM \( { \mathbf{Z}_0 } \), associated to the initial core set, is non-singular by construction. Therefore, the first right-hand term is just the solution of the FQQSPR linear equation for the initial core set , as shown in Eq. (200). Thus, calling:
then, Eq. (203) could be rewritten as any of the two following equalities:
Taking into account Eq. (205), the second component can be written as:
where:
expression (207) shows the expected trivial result consisting of how the U-m property and the coefficient, still not evaluated, obviously are linearly related.
Analysis of the FQQSPR Equation.
Only in the case that the U-m property can be associated to a concrete numerical value: \( { \pi _U } \), say; then the exact linear coefficient: \( { x_U } \) can be written in terms of the quantities appearing in Eq. (206) as:
Equation (208), although it will never hold exactly by construction, tells us about several interesting features.
First, admitting through the previous discussion that the vector \( { |\mathbf{q}\rangle } \) is nothing else than the exact linear coefficient set for the core set, then the scalar product: \( { a_0 = \langle\mathbf{z}|\mathbf{q}\rangle \equiv \pi^{(0)} } \), is nothing else than an estimation of the U-m property, \( { \pi^{(0)} } \), say, using the discrete representation of the U-m with respect to the core set. Thus, the numerator of Eq. (208) corresponds to the difference between this rough approximation and the exact property value, if known, of the U-m. Obviously enough, if: \( { \pi_U = \pi^{(0)} } \) holds, then the U-m coefficient will be null, as the U-m property could be solely computed by the descriptor \( { |\mathbf{z}\rangle } \).
Second, the denominator in Eq. (208), tells us about the difference between U-m self-similarity and the norm of the vector \( { |\mathbf{z}\rangle } \) computed in the QSM reciprocal space, defined as the vector space where the inverse of the QSM acts as a metric matrix. An identity as: \( { \theta = \langle\mathbf{z}|\mathbf{Z}^{-1}|\mathbf{z}\rangle } \), will produce an unacceptable linear algebra result, whichever value the U-m property could be. It is plausible to suppose, therefore, that in well-behaved FQQSPR prediction problems, the following inequality shall always hold: \( { \theta \ne \langle\mathbf{z}|\mathbf{Z}^{-1}|\mathbf{z}\rangle } \).
One can conclude, within the settings of the U-m prediction problem, that the linear structure of the FQQSPR equation does not permit the evaluation of the U-m property in Eq. (206), unless the coefficient x appears defined in some way. The exact coefficient value x U can be derived, if and only if, a given concrete value of the property is known, but in this case, the prediction problem will not a priori exist as such. Only if the property of the U-m appears in a parametrized form, the problem can be handled in an approximate way.
Thus, one arrives at the logical conclusion in that a prediction obstacle is already present in the case of a one-dimensional representation of the QSPR operator O, even if the quantum similarity description discrete vector tag: \( { \begin{pmatrix}|\mathbf{z}\rangle \\ \theta\end{pmatrix} } \) is known for the extra added U-m structure, but the corresponding property value is not defined, but considered as a parameter.
Analysis of U-m Predicted Property Values.
Therefore, the aim of the following discussion will be to find an appropriate way to determine a reasonable optimal approach for the U-m coefficient x, by means of manipulating Eq. (205) in order that the unknown property parametrized value p could be estimated using Eq. (206). If some optimal coefficient value \( { x^{(\text{opt})} } \) is found, Eq. (206) can be rewritten as:
in this way the role of the estimated coefficient \( { x^{(\text{opt})} } \) appears with a clear meaning now: it constitutes one of the factors to correct the rough initial estimate of the U-m property \( { \pi^{(0)} } \), which can be obtained from the primary information provided by the core set by just using Eq. (201). Equation (209) above also enhances the leading role of the U-m self-similarity ? for such a property correction-estimation task. In Eq. (209), the U-m self-similarity appears shifted, in turn, by the norm of the U-m discrete representation vector, \( { |\mathbf{z}\rangle } \), with respect to the core set: \( { \langle\mathbf{z}|\mathbf{a}\rangle = \langle\mathbf{z}|\mathbf{Z}^{-1}|\mathbf{z}\rangle } \), computed over reciprocal space.
Formulation of the Optimization Problem
In any of both direct and reciprocal space cases, as expected from the linear structure of the fundamental equations used and provided that: \( { \lambda \in \mathbf{R} } \), then it can be written for the unknown sought property:
also, the equation for the core set unknowns can be written in general as:
where A is a positive definite matrix.
The unknown property in Eq. (210) will be well-defined whenever, using Eq. (211), one could obtain a well-defined value of the parameter: ?. As the solution of Eq. (210) corresponds to an infinite collection of real elements, the restricted solution in the case of putting one molecule in, is not unique, as from Eq. (211) one can describe several possible ways to obtain optimal values of the parameter ?. For instance:
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a)
Defining the difference vector: \( { |d\rangle = |\pi\rangle -\lambda|a\rangle } \), a difference norm can be constructed:
$$ \langle d|d\rangle = \langle\pi|\pi\rangle -2\lambda\langle\pi|a\rangle +\lambda^2\langle a|a\rangle\:, $$(212)optimizing the expression (212) with respect to the parameter, provides:
$$ \lambda^{\text{opt}} = \frac{\langle\pi|a\rangle}{\langle a|a\rangle}\:, $$besides the optimal value of the difference norm will be a minimum, as the second-order coefficient in Eq. (212) is a Euclidean norm of a non-null vector.
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b)
One can consider the norm of vector \( { |\mathbf{u}\rangle } \) as defined in Eq. (211) the objective function to be optimized; in this case it can be written:
$$ \langle \mathbf{u}|\mathbf{u}\rangle = \langle\pi|\mathbf{A}|\pi\rangle -2\lambda\langle\pi|\mathbf{A}|a\rangle +\lambda^2\langle a|\mathbf{A}|a\rangle\:, $$So the optimal value of the parameter is now:
$$ \lambda^{\text{opt}} = \frac{\langle\pi|\mathbf{A}|a\rangle}{\langle a |\mathbf{A}|a\rangle}\:, $$which provides a similar form as in the previous procedure, weighted by the transformation matrix A.
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c)
The scalar product of the vectors \( { \{\mathbf{u}\rangle ;|\pi\rangle\} } \) can be optimized, the objective function is now:
$$ \begin{aligned} |\langle\pi|\mathbf{u}\rangle|^2 &= |\langle\pi|\mathbf{A}|\pi\rangle - \lambda\langle\pi|\mathbf{A}|a\rangle|^2 \\ &=|\langle\pi|\mathbf{A}|\pi\rangle|^2 -2\lambda\langle\pi|\mathbf{A}|\pi\rangle \langle\pi|\mathbf{A}|a\rangle\\& \quad + \lambda^2|\langle a|\mathbf{T}|a\rangle|^2\:, \end{aligned} $$producing:
$$ \lambda^{\text{opt}} = \frac{\langle\pi|\mathbf{A}|\pi\rangle}{\langle\pi|\mathbf{A}|a\rangle}\:. $$ -
d)
The scalar product of the vectors \( { \{\mathbf{u}\rangle;| a\rangle\} } \) can be now optimized, in an equivalent way as in the previous procedure; that is, using the objective function:
$$ \begin{aligned} |\langle t|\mathbf{u}\rangle|^2 &= |\langle a|\mathbf{A}|\pi\rangle -\lambda\langle a|\mathbf{A}|a\rangle|^2 \\ &= |\langle a|\mathbf{A}|\pi\rangle|^2 -2\lambda\langle a|\mathbf{A}|\pi\rangle\langle a|\mathbf{A}|a\rangle\\&\quad +\lambda^2|\langle a|\mathbf{A}|a\rangle|^2\:, \end{aligned} $$which permits one to obtain the optimal value:
$$ \lambda^{\text{opt}} = \frac{\langle a|\mathbf{A}|\pi\rangle}{\langle a|\mathbf{A}|a\rangle}\:; $$this result, however, corresponds to the samerestriction as the one previously studied in procedure II of reference [60]. Thus, optimizing the norm \( { \langle\mathbf{u}|\mathbf{u}\rangle } \) seems to be equivalent to optimizing the squared module: \( { |\langle a|\mathbf{u}\rangle|^2 } \).
A Quadratic Error Restricted First-Order (n+1) Estimation
The easiest procedure to overcome the previously mentioned evaluation impasse for the unknown property of the U-m, concretely the approximate evaluation of the coefficient x, appears naturally associated to the possibility to introduce a restriction of some sort into the FQQSPR equation solution. Here follows the description of one among some possible restriction procedures. One will discuss a second option and sketch some alternative procedures as well, within a separate section below.
Setting up the Problem.
By inspecting Eq. (205), one can define the difference vector:
and compute with it the following associated quadratic error, which in this case describes a second-order polynomial of the unknown coefficient x:
which, in turn, using the usual null gradient condition, allows us to obtain an optimal value of the coefficient x, obeying the simple quotient expression:
The optimal coefficient value (215) produces a minimum of the quadratic error, being the second-order coefficient of the quadratic error polynomial (214), associated to the positive definite Euclidean norm of the U-m discrete representation with respect to the core set density tags, that is: \( { \langle\mathbf{z}|\mathbf{z}\rangle > 0 } \). Such a quadratic error restriction is equivalent to constructing a difference vector (213) with elements as small as possible. In the case, quite unlikely to occur, where the known property vector \( { |\mathbf{p}_\mathbf{0}\rangle } \) and the U-m quantum similarity vector \( { |\mathbf{z}\rangle } \) are linearly dependent, the present restriction will construct a difference vector (213), which will be exactly the null vector at the optimal value of the unknown.
Using \( { x^{(\text{opt})} } \), the optimal value obtained with Eq. (215), the corresponding unknown property for the U-m can be straightforwardly predicted using Eq. (206):
providing an expression, which can be easily rearranged by defining the matrix:
Therefore, one can compactly write the optimal property equation for the U-m as:
moreover, defining the \( { (1\times n) } \) row vector: \( { \langle\mathbf{b}| = \langle\mathbf{z}\mathbf{A} } \); then, this new description permits us to write the estimated property by means of the following scalar product form:
Hence, within the linear FQQSPR equation under the minimal quadratic error restriction, the result (216) shows that the estimated optimal unknown property for any U-m, is always expressible as a linear functional of the known molecular properties of the core set. Such a result is in agreement with usual classical QSPR treatments.
Additionally, in a very unlikely case, where a linear dependence of the core set property vector \( { |\mathbf{p}_\mathbf{0}\rangle } \) and the U-m quantum similarity vector: \( { |\mathbf{z}\rangle } \) applies, that is whenever: \( { |\mathbf{p}_0\rangle = \lambda|\mathbf{z}\rangle } \); then, the optimal estimated property value will be expressible as a multiple of the U-m self-similarity:
It must be finally noted in any case that the gauge operator expectation value: \( { \langle\sigma[\rho_U]\rangle } \), for the U-m, if different from zero, shall be added to the optimal value of the property in Eq. (216) or (217) in order to retrieve the predicted value corresponding to the original property set.
A (\( { 2+1 } \)) Naïve Application Example.
In order to illustrate the above procedure one can consider a simple case as follows. Supposing that the core set is made of just two molecular structures: \( { \{A,B\} } \) say, with a known shifted property vector:
and also admitting that the U-m, possessing the unknown parametric property, p, could be labeled as \( { \{U\} } \); then, the QSM of the core set and the similarity vector of the U-m can be respectively written as:
with \( { z_{UU} } \) representing the U-m self-similarity measure. One can readily compute the core set similarity matrix inverse:
It must be now said that when doing this kind of calculation care must be taken with the values of the \( { (2\times 2) } \) similarity matrix determinant D, because a value approaching zero can render the procedure useless and generate unpredictable computational errors. For all molecular pairs \( { \{A,B\} } \) of the core set , the value of the determinant D has to be checked to be significantly greater than a positive definite threshold, that is:
failure to comply with this condition for any core set molecular pair may well represent a computationally unbalanced QSM triplet \( { \{A,B,U\} } \). This test shall be added to the already described coherent calculation procedures, when accurate QSM have to be computed.
However, the positive definite determinant condition can also be rewritten in a positive definite quantum similarity matrix condition, that is:
A \( { (N\times N) } \) positive definite condition problem, which corresponds in general to the positive definite nature of the quantum similarity matrices, can be shown that it can be readily solved for any core set cardinality, but the nature of this subject, although of capital importance for application purposes, appears to be marginal in the present work and hence will be studied elsewhere.
Thus, one can express the needed vector resulting from the product: \( { \mathbf{Z}_0^{-1} |\mathbf{z}\rangle } \) as:
and the same can be obtained for the scalar products entering the restricted optimal solution:
Finally, one can also write:
and
Therefore, one can obtain the unknown optimal property value, after some trivial manipulation of the previous quantities as:
where the following symbols are used:
The expression (218) for the optimal quadratic error restricted property of the U-m self-similarity, constitutes an explicit example involving a very limited number of molecular structures. However, it also corresponds to a general equation involving any triad of molecules, where one of them acts as the U-m.
This simple way of estimating a property can be structured into a procedure involving all the: \( N=1/2[n(n-1)] \) possible core set distinct molecular pairs. Indeed, given a core set and a U-m, one can compute all the possible property estimates using Eq. (218). Such a process will produce a set of N values of the U-m estimated property:
which can be finally manipulated in the usual statistical way.
Such an example opens the way to other possible choices using as probe core sets three or another number of QO. In order to leave this study within reasonable limits this possibility will not be further investigated here.
The (\( { n+m } \)) Case Under a Quadratic Error Restriction.
One can extend the estimation procedure, outlined in the previous section, in order to include a U-m set of arbitrary cardinality, m say, so a general quadratic error restricted scheme can be also described in this more general case. One may write the partition of the QSM into the core set, bearing the label 0 and the U-m set, bearing the label 1, then the FQQSPR equation can be written as:
which produce the two matrix equations, as follows:
So, from the first element of Eq. (220), one can deduce:
with the possibility to construct a difference vector:
Then, one may immediately use the difference vector (221) to define a quadratic error function like:
which upon derivation and submitted to the extremum condition of null gradient, produces:
so, the U-m set unknown coefficients, restricted to minimal quadratic error, can be obtained by means of the matrix expression:
The solution in Eq. (223) depends only on the circumstance that the matrix:
shall be non-singular. In fact, the matrix \( { \mathbf{A}_{11} } \) corresponds to the scalar products of the matrix representations of the U-m set with respect to the core set elements. This condition, if the computed similarities are not faulty, constitutes a metric matrix of the U-m space, subtended by the U-m QSM columns. Thus, provided that the U-m discrete representations with respect to the core set are linearly independent, the inverse of A is guaranteed to exist, as it will be positive definite; moreover, implying the quadratic error (222) is a minimum at the value \( { \big|\mathbf{x}_1^{\text{Opt}}\big\rangle } \) given by Eq. (223).
One can easily estimate the unknown parametrized U-m property vector \( { |\mathbf{p}_1\rangle } \), submitted to the quadratic error restriction, after defining the auxiliary matrix:
and performing some rearrangements using trivial matrix algebra, it is obtained:
Equation (224) shows that the predicted U-m set property vector is a linear transformation of the core set known properties, a result consistent with the unique U-m case, already described in the previous section and coincident with the usual classical QSPR procedures.
Alternative Restrictions and the Associated Prediction Algorithms
The case analyzed in Sect. “Formulation of the Optimization Problem” is not at all unique. One can describe other possible alternative restrictions, which can be imposed to the FQQSPR equation, as follows in this section.
An Alternative Orthogonality Restriction.
Here, choosing one of the possible procedures, a deep discussion will be carried out for a U-m set bearing one element only, because the extension to the case of several elements is trivial and similar to theprevious Subsect. “A Quadratic Error Restricted First-Order (n+1) Estimation” development, although a brief outline will be given for the sake of completeness. Finally, the remnant plausible restrictions will be only sketched, because the procedure to obtain the application algorithms follows the same trends as in the explicit examples.
-
a)
The \( { (n+1) } \) framework
For the purpose of finding an alternative restriction to the one described in the previous Subsect. “A Quadratic Error Restricted First-Order (n+1) Estimation”, it is necessary to recover the first matrix equation of the partition (202), leading to Eq. (203). Then, upon left multiplying both sides by the row vector \( { \langle\mathbf{z}| } \), one can obtain:
$$ \langle\mathbf{z}|\mathbf{x}_\mathbf{0}\rangle = \langle\mathbf{z}|\mathbf{Z}_\mathbf{0}^{-\mathbf{1}}|\mathbf{p}_\mathbf{0}\rangle -x\langle\mathbf{z}|\mathbf{Z}_\mathbf{0}^{-\mathbf{1}}|\mathbf{z}\rangle =\alpha _0 -\alpha_1 x\:. $$(225)Then, one can use the resulting Eq. (225) to minimize the scalar product \( { \langle\mathbf{z}|\mathbf{x}_\mathbf{0}\rangle } \), appearing on the left-hand side. As in the previous treatment, the right-hand side of Eq. (225) can be considered a difference, which can generate a quadratic error function to be minimized with respect to the unknown parameter, x, which can be evaluated in this manner afterwards. After a trivial manipulation one finds:
$$ \varepsilon^{(2)} = |\langle\mathbf{z}|\mathbf{x}_\mathbf{0}\rangle|^2 = |\alpha_0 -\alpha_1 x|^2 = \alpha_0^2 -2\alpha_0 \alpha_1 x+\alpha_1^2 x^2\:, $$(226)and in this way the extremum condition imposed upon Eq. (226), becomes a manner to obtain the optimal value of the unknown coefficient:
$$ 2\alpha_0 \alpha_1 -2\alpha_1^2 x = 0 \to x_\bot^{\text{opt}} = \frac{\alpha_0}{\alpha_1} = \frac{\langle\mathbf{z}|\mathbf{Z}_\mathbf{0}^{-\mathbf{1}}|\mathbf{p}_\mathbf{0}\rangle}{\langle\mathbf{z}|\mathbf{Z}_\mathbf{0}^{-\mathbf{1}}|\mathbf{z}\rangle}\:. $$(227)The result depicted in the quotient of expression (227), turns out to be equivalent to agreeing that the imposed restriction considers the vectors orthogonal in the scalar product (225), or: \( { \langle\mathbf{z}|\mathbf{x}_\mathbf{0}\rangle = 0 } \). Admitting that, such a restriction produces a second equation in the system (202), which simplifies to:
$$ \pi_\bot^{\text{opt}} = \theta x_\bot^{\text{opt}} = \theta \frac{\langle\mathbf{z}|\mathbf{Z}_\mathbf{0}^{-\mathbf{1}}|\mathbf{p}_\mathbf{0}\rangle}{\langle\mathbf{z}|\mathbf{Z}_\mathbf{0}^{-\mathbf{1}}|\mathbf{z}\rangle}= \theta\frac{\langle\mathbf{z}|\mathbf{q}\rangle}{\langle\mathbf{z}|\mathbf{a}\rangle}\:. $$(228)Expression (228) resembles the limiting case value of the unknown property: \( { \pi_\bot^{\text{opt}} } \), for the quadratic error restriction studied in Subsect. “A Quadratic Error Restricted First-Order (n+1) Estimation”, as shown into Eq. (217), when the vector of the known properties and the U-m discrete representation with respect to the core set become linearly dependent. Similar scalar products appear in both expressions. However, in the present orthogonal restriction, they are evaluated using as a metric the inverse of a matrix, which is nothing else than the QSM, associated to the core set space. Thus, the scalar products entering Eq. (228) are computed in the reciprocal space of the vector space possessing a metric \( { \mathbf{Z}_0 } \), as previously commented in Sect. “Linear Quantum QSPR Fundamental Equation”. Moreover, the approximate coefficient value in Eq. (227) resembles the exact expression of the U-m coefficient, as described in Eq. (208). Rewriting Eq. (228) as:
$$ \pi_\bot^{\text{opt}} = \bigg(\frac{\theta}{\langle\mathbf{z}|\mathbf{a}\rangle}\bigg)\pi^{(0)} = \omega\pi^{(0)}\:. $$It can be easily seen how the ratio between the U-m self-similarity and the norm of the U-m representation with respect to the core set in reciprocal space, corrects in this case the rough estimate of the U-m property.
-
b)
An alternative restriction case
A variant of the restriction discussed up to now can be easily described. Instead to optimize the scalar product: \( { \langle\mathbf{z}|\mathbf{x}_\mathbf{0}\rangle } \), the alternative scalar product: \( { \langle\mathbf{p}_\mathbf{0}|\mathbf{x}_\mathbf{0}\rangle } \) can be minimized, so the optimal coefficient will be given by:
$$ x^{\text{opt}} = \frac{\langle\mathbf{p}_\mathbf{0}|\mathbf{Z}_\mathbf{0}^{-\mathbf{1}}|\mathbf{p}_\mathbf{0}\rangle}{\langle\mathbf{p}_\mathbf{0}|\mathbf{Z}_\mathbf{0}^{-\mathbf{1}}|\mathbf{z}\rangle}\:. $$(229) -
c)
The simplified \( { (2+1) } \) framework as a naïve application example
The simplified situation (\( { 2+1 } \)), concerning three molecules, as in the former case of Subsect. “Extended Wave and Density Functions”, is simple to solve, for the previous resultant equation in Section 6.1.1, as the involved elements have already been described, when the quadratic error restriction (214) was studied. Under the present orthogonal restriction and using the same notation as the one appearing in the former discussion in Subsect. “Extended Wave and Density Functions”, now one can express the estimated property of the U-m as the quotient:
$$ \pi_{\bot,AB;U}^{\text{opt}} = z_{UU} \frac{\left[\begin{array}{c} p_A(z_{BB} z_{AU} -z_{AB} z_{BU})\\ +p_B (z_{AA} z_{BU} -z_{AB} z_{AU})\end{array}\right]}{\left[\begin{array}{c}z_{AU}(z_{BB} z_{AU} -z_{AB} z_{BU})\\ + z_{BU} (z_{AA} z_{BU} -z_{AB} z_{AU})\end{array}\right]}\:. $$An equivalent statistical procedure involving the core set pairs, as the one described at the end of Subsect. “Extended Wave and Density Functions”, can be obviously followed in this case too.
-
d)
The \( { (n+m) } \) framework
The \( { (n+m) } \) algorithm can be also easily set, employing the partition (219) and also Eqs. (220), thus generalizing the algorithm described in Subsect. “Evaluation of Unknown Molecular Properties as Expectation Values”. The first equation multiplied by the matrix \( { \mathbf{Z}_{01}^\mathrm{T} } \) on the left in both sides of the equality sign, provides:
$$ \mathbf{Z}_{01}^\mathrm{T} |\mathbf{x}_0\rangle = \mathbf{Z}_{01}^\mathrm{T} \mathbf{Z}_{00}^{-1} [|\mathbf{p}_0\rangle - \mathbf{Z}_{01}|\mathbf{x}_1\rangle]\:, $$and the restriction:
$$ \mathbf{Z}_{01}^\mathrm{T} |\mathbf{x}_0\rangle = |\mathbf{0}\rangle $$corresponds to considering the vector \( { |\mathbf{x}_0\rangle } \) as a member of the null space of the matrix \( { \mathbf{Z}_{01}^\mathrm{T} } \). Such equality permits, in turn, to write the optimal vector \( { |\mathbf{x}_1\rangle } \) as:
$$ \big|\mathbf{x}_{\bot;1}^{\text{Opt}}\big\rangle = \big(\mathbf{Z}_{01}^\mathrm{T} \mathbf{Z}_{00}^{-1} \mathbf{Z}_{01}\big)^{-1}\big(\mathbf{Z}_{01}^\mathrm{T} \mathbf{Z}_{00}^{-1}\big)|\mathbf{p}_0\rangle\:, $$so, the estimated property vector under the null space restriction simply becomes:
$$ |\mathbf{p}_{\bot;1}^{\text{Opt}}\rangle = \mathbf{Z}_{11} |\mathbf{x}_{\bot;1}^{\text{Opt}}\rangle\:, $$As in the former quadratic error restriction discussed in Subsect. “Expectation Values Within Extended Density Functions Framework”, one obtains an expression which shows that this result is nothing else than a linear transformation of the vector of the core set properties.
Other Possible Restriction Choices.
Besides the previously discussed pair of alternative restriction choices and the one outlined in Section 6.1.2, one can describe other possible procedures to compute the optimal U-m coefficient. They will be briefly explained for a U-m set bearing one element only, as their generalization to \( { (n+m) } \) situations and the \( { (2+1) } \) simplification could be done using the same procedures as before.
-
a)
Quadratic error in reciprocal space vectors: \( { |\mathbf{x}_0\rangle } \) norm restriction
First, one can recall for this purpose Eq. (204), then the core set coefficient vector \( { |\mathbf{x}_0\rangle } \) may be expressed with the two basic vectors \( { \{|\mathbf{p}_0\rangle,|\mathbf{z}_0\rangle\} } \) transformed into the reciprocal space: \( { \{|\mathbf{q}\rangle,|\mathbf{a}\rangle\} } \):
$$ |\mathbf{x}_0\rangle = |\mathbf{q}\rangle -x|\mathbf{a}\rangle\:. $$(230)Now, the Euclidean norm of the vector \( { |\mathbf{x}_0\rangle } \) can be optimized, providing the optimal U-m coefficient as:
$$ x^{\text{Opt}} = \frac{\langle\mathbf{q}|\mathbf{a}\rangle}{\langle\mathbf{a}|\mathbf{a}\rangle} = \frac{\langle\mathbf{p}_0|\mathbf{Z}_0^{-2}|\mathbf{z}\rangle}{\langle\mathbf{z}|\mathbf{Z}_0^{-2}|\mathbf{z}\rangle}\:, $$(231)which is a variant of the form obtained in Eq. (215). The \( { (n+m) } \) case can be easily handled as within the previous discussion on the two described restrictions.
-
b)
Several alternative plausible restrictions within reciprocal space
Finally, another possible restriction set must be described, which can be associated to Eq. (230). Instead of minimizing the norm of the coefficient vector \( { |\mathbf{x}_0\rangle } \) one can minimize either the scalar product \( { \langle\mathbf{q}|\mathbf{x}_0\rangle } \), or work on the alternative form \( { \langle\mathbf{a}|\mathbf{x}_0\rangle } \). The first option provides the optimal U-m coefficient:
$$ x^{\text{Opt}} = \frac{\langle\mathbf{q}|\mathbf{q}\rangle}{\langle\mathbf{q}|\mathbf{a}\rangle}\:, $$which constitutes an expression related to Eq. (229), while the form deduced from the second one is equivalent to Eq. (231).
Some Application Remarks.
The first-order FQQSPR equation does not possess immediate predictive power. In order to circumvent this limitation though, one can easily show that two alternative approximate algorithms may be employed, among other possible similar choices. These procedures can be used to estimate the unknown properties of one or various molecules described as QO.
The present algorithms produce similar formal structures, which can be easily connected with classical QSPR points of view. Such a resemblance can be also simply used to manipulate similar, but empirical, equations in the classical QSPR framework, where the computational formalism appears to be of the same characteristics as in linear QQSPR problems. In order to use the algorithms described here in empirical QSPR cases, there is only need to substitute the QSM, which is the basic matrix in QQSPR procedures, by the Gramian matrix of the molecular descriptor set as defined in Eq. (198), which is the comparable molecular space matrix which can be constructed in classical QSPR. In an indirect manner, therefore, the present study provides an alternative to the widespread QSPR algorithms based on the space descriptor path, a new classical QSPR procedure, which appears, from now on, to be accompanied by a quite diverse toolbox set, common to the linear QQSPR framework, in order to obtain predictions of unknown properties in empirical studies.
Finally, the present results, although exhaustive as far as one can see but without discarding the existence of alternative FQQSPR equation restrictions, from the theoretical point of view they lie on the linear QQSPR framework, they have thus to be considered just as a first step in order to generally solve the prediction problem in QQSPR. This is so as, contrary to classical QSPR procedures, the extension of the FQQSPR equation to higher order terms can be easily described, as well as employed within a set of similar ideas and procedures as these herein discussed.
Extensive numerical results and additional study of high order level problems seems therefore to outline the future research in the open QQSPR area of study.
One Molecule at a Time Linear QQSPR
When constructing the linear QQSPR equation one can choose a system of one core molecule and one U-m, which will constitute the simpler case. The similarity matrix is:
Where the subindex I stands for any molecule in the core set, that is: a well defined molecular structure with a known property p I and U for any well-defined molecule with an unknown propertyp, which has to be estimated. It can be written:
Taking the first equation and substituting into the second:
an expression which, after rearrangement, provides a way to estimate the unknown property:
where \( { \Delta =\operatorname{Det}(\mathbf{Z}) } \).
One can see the undetermined coefficient c U as equivalent to a parameter ? which in turn can be optimized, thus the unknown property could be rewritten as:
There are several ways to obtain the optimal value of the parameter ?, but all of them are equivalent. For example, one can try to make optimal the coefficient c I in the first equation:
so, in this way the optimal U-m property is:
Then the problem consists of obtaining the U-m self-similarity and the similarity between the core molecule and the U-m. So, for every core set C molecular structure one can obtain an estimate of the U-m property, say:
Then, supposing that the cardinality of the core set is \( { N\colon\# C=N } \), one can obtain a statistical average of all the core set estimates of the U-m property:
An expression which proves that this simple QQSPR formulation arrives at the usual QSPR result, consisting of the fact that the estimated value of the property of the U-m is a weighted sum of the properties of the core set:
the weights being simply the ratios between the U-m quantum self-similarity and the quantum similarity measure of the U-m with every core set molecular structure.
Practical Considerations.
However, in this or other more sophisticated cases, the estimation procedure can be achieved in two steps. First, the elements of the core set can be employedas the U-m ones, one by one in front of the remnant \( { N-1 }\) in a kind of Leave One Out procedure. The N optimal estimated values, \( { \{\langle\pi_I\rangle\} } \) say, in this way can be fitted to the experimental property ones, providing in this manner a simple, Hansch-like relationship :
a relationship which can be further employed to estimate the experimental values of the U-m elements \( { \{p_U\} } \), by using the above-defined equation:
Moreover, an interesting feature of this procedure is that each estimated value, obtained through solving the fundamental QQSPR equation and irrespective of the fact that the estimation is made over the C or U set elements, can be associated to a mean value, obtained over the set of C-m and also attached to a variance. It is a simple matter of elementary statistical theory application to obtain confidence limits for each estimate, and thus to gather information about, for instance, the outlier nature of some elements and the goodness-of-fit of the whole procedure.
One Molecule at a Time: Quadratic Terms in QQSPR
The operator which can be employed as the source of the fundamental QQSPR equation may be expressed with quadratic and superior terms, within a sequence involving the density elements of the C-m and U-m elements:
The pair of expectation values of both molecules can be easily written up to third order as:
and
where use has been made of the similarity measures like:
and triple similarity measures , for instance:
the properties as expectation values can be rewritten employing the ket-matrix notation:
with a similar notation for the ket \( { |\mathbf{z}_U\rangle } \) and the matrix \( { \mathbf{Z}_U } \); the bra notation signifying the corresponding ket transposes. Therefore:
The QQSPR problem consists of the fact that the coefficient vector is not only unknown \( { |\mathbf{w}\rangle } \) but also the U-m property p. In fact, the quadratic system which corresponds to the quadratic fundamental QQSPR equation in this case, is a set of two different quadratic functions of the same two variables: \( { |\mathbf{w}\rangle } \). The solution may be more complicated than in the linear case, but the procedure can be described in similar terms. That is, first use the C-m equation to express the coefficient w I in terms of the U-m coefficient and the corresponding similarity measures elements. Then optimize such a coefficient with respect to the U-m one, considered as a parameter. The optimal values of w U can be employed to evaluate an optimal value of the unknown U-m property p.
A Possible Algorithm.
The first fundamental QQSPR equation can be easily transformed into the second-order polynomial root seeking structure.
which provides:
and after rearranging the square root part:
the coefficient w I appears in terms of w U and the implied similarity integrals. Also, this expression can be employed in the second fundamental QQSPR equation to obtain the U-m property in terms of only one parameter. As in the linear case, the expression of w I can be optimized with respect to w U , which can be considered now as a parameter. The expression to be optimized can be written as:
Thus, the equation yielding the optimal value of w U can be easily written as:
yielding:
this value permits us to compute \( { w_I^{\text{Opt}} } \) by means of Eq. (235) and therefore \( { \pi ^{\text{Opt}} } \) can be obtained with Eqs. (235) using the original form (233).
Alternative Unrestricted Variational Algorithm.
Starting again from the quadratic Eq. (234), one can vary both parts of the FQQSPR equation:
so, the optimal estimate values of the C-m and U-m properties will be given by:
which can be associated to minimal values, as the second-order similarity matrices are constructed to be positive definite and thus:
So, an ultimate procedure could be designed, starting to obtain with every one of the core set elements the following linear equation via a least squares procedure:
in such a way that the linear equation above provides the possibility to obtain the final estimate of the U-m property value:
Notes
- 1.
The Hadamard product (sometimes also called Schur or Kronecker product) is related to the multiplication result of two sums and constructed by the sum of the resultant diagonal cross-terms only. In this way, the inward (or Hadamard) product of two sums can be specified by the following algorithm:
$$ \bigg(\sum\limits_I^N a_I\bigg)\ast \bigg(\sum\limits_I^N b_I\bigg) = \sum\limits_I^N a_I b_I\:. $$Both sums shall have the same number of terms N, for the IMP being feasible.
- 2.
By the symbol: \( { \mathbf{A}\ast > 0 } \), applied to an arbitrary matrix \( { \mathbf{A}=\{a_{ij}\} } \), is meant that \( { \forall i,j\colon a_{ij} > 0 } \).
- 3.
In the case where the nature of the operator or the set of operators, active in the Sobolev norm definition, has to be specified, then the notation Hilbert–Sobolev ?-, O- or ?-spaces, can be obviously employed.
Abbreviations
- Glossary:
-
There is a brief description of the terms used herein. The defined items appear in alphabetical order. When in a definition a term already defined in the glossary is mentioned, it is written in bold face; then, the reader has to refer to the corresponding glossary item, where more information is given.
- Carbó (similarity) index:
-
The Carbó (Similarity) Index is a QS index, which corresponds to the cosine of the angle subtended by the density function (DF) tags of any pair of quantum objects. The values of the Carbó index lie within the interval \( { (0,1] } \). The lower bound corresponds to a complete dissimilarity, while the unit value is encountered when comparing a quantum object with itself. See also: Euclidian Distance (Similarity) Index.
- Core set:
-
A QOS with the additional information of a known and well-defined property value for every QO.
- (First order) density function (DF):
-
The first order DF is a quantum theory concept, associated to a known electronic submicroscopic system. To be understood by this term is a wave function squared module or the full DF, reduced by integration over the space and spin coordinates to a function of the space coordinates of a unique electron. As it is customary in the literature, the name will be shortened to DF. Such a function can be also obtained via direct computation within Density Functional Theory (DFT). DF can be employed as tags associated to well-defined quantum objects. DF are non-negative functions. According to the usual quantum mechanical interpretation, within the DF collection there is contained all the information one can extract from the system. This last statement is the basic postulate which appears in first place when developing QQSPR theory.
- (Full) density function:
-
A quantum theoretical concept, associated to a known electronic (or other particle sets) system. It corresponds to the system's wave function squared module.
- Density function (DF) tags:
-
The tag set, (see: tagged set) collecting the quantum mechanical density functions used as descriptors in a QOS.
- DQOS:
-
Discrete Quantum Object Set. A QOS whose tags are finite dimensional vectors, whose elements, in turn, are computed as quantum similarity measures.
- Euclidean distance (similarity) index:
-
Like the Carbó Index the Euclidean Distance Index is a quantum similarity index, involving the Euclidean distance between two density functions. The range of this index is \( { [0,\infty) } \), thus their minimal values correspond to two identical quantum objects, while the index grows in relation to the difference in the compared quantum objects.
- FQQSPR equation:
-
Fundamental Quantum Quantitative Structure-Properties Relationships equation. The subject of study and analysis of the present contribution. A non-empirical equation, which can be deduced by quantum mechanical theoretical means, serving as the basic tool to obtain QQSAR or Quantum QSAR. Here, only the linear FQQSPR equations are deeply discussed, as they are the main source of QQSPR studies, but the FQQSPR equation can easily be extended to any order.
- Molecular descriptors:
-
Parameters of varied origin: empirical, theoretical or experimental, or functions obtained from quantum mechanical manipulations like the Density Function or the Electrostatic Molecular Potential, or by empirical considerations, associated to a given molecule, which represent the molecular structure environment and can be employed to obtain QSPR. Any set of parameters or functions, which can be used as tags in a tagged set, whose objects are molecules.
- QO:
-
Quantum Object. An element of a QOS, a tagged set where every element is constructed as a composite of a submicroscopic system description (the object) and a density function (the tag). The possibility to represent all the information contained within any quantum system by the full or reduced DF, permits one to describe such an entity formed by the structure of the system and its DF together; this can be called a QO. Plural: QO's.
- QOS:
-
Quantum Object Set. A tagged set made of QO's.
- QS:
-
Quantum Similarity: A discipline dealing with similarity measures between submicroscopic systems. Such kinds of quantum similarity measures (QSM) can be computed using the quantum theoretical description of such kinds of objects. According to quantum theory all the information one can obtain about a quantum system is contained in the state DF and the set of possible reduced forms, hence quantum similarity is part of this information contained in the DF. Usually, for computational convenience, the (first order reduced) DF has been employed as a universal descriptor for comparison purposes and it is the employed tag of a QO. Similarity comparisons become possible by means of comparing the QO DF tags.
- QSAC:
-
Quantum Similarity Aufbau Condition: A condition that the QS Matrix has to comply with in order to be positive definite and admit Choleski decomposition.
- QSAR & QSTR:
-
QSAR are QSPR employed to estimate biological molecular activity values via molecular descriptors. When QSPR are employed to estimate molecular toxicity, they can be called QSTR.
- Quantum similarity matrices (QS matrices):
-
[Depending on the context so as not to cause confusion with the term QS Measures the abbreviation QSM can also be used to denote QS Matrices]: Any matrix which contains computed QSM results involving several QO. Usually the QS Matrix is symmetric and square when the QSM on the involved elements of a unique QOS are ordered in pairs. In this case the QSAC must hold. The QS Matrix can be rectangular when computing and ordering into a matrix the QSM of two different QOS. QS Hypermatrices appear when higher order QSM associated to more than two DF are involved.
- Quantum similarity measures (QSM) and indices (QSI):
-
A positive definite integral, computed employing the QO DF tags as integrands and, if necessary, including a positive definite operator which can be supposed to act as a weight. Possessing the structure of a measure, such an integral can be interpreted as a generalized volume. QSM between two or several QO correspond to integrals, which can be constructed primarily with integrands made with the product of the density tags of the compared quantum objects plus a positive definite operator. Such a definition ensures the positive definite nature of the QS integrals. A quantum self-similarity measure corresponds to the QSM computed with the density function tags of a unique QO. QSI are manipulations of the QSM in order to obtain QS comparisons within an adequate range.
- QSPR:
-
(Classical) Quantitative Structure-Properties Relationships. This term refers to any empirical relationship permitting the connection between molecular structure, represented by a set of parameters (molecular descriptors) of any origin and molecular properties. Usually, by a classical QSPR can be understood a non-causal multilinear relationship, obtained via statistical reasoning and procedures. This is the generic name given to any functional (usually linear) connecting the properties of a molecule with the attached molecular descriptors. The QSPR functionals are empirically obtained by statistical analysis, usually employing (non-)linear regression techniques or any variant of it.
- QQSPR:
-
Quantum Quantitative Structure-Properties Relationships. These are non-empirical functionals derived from the structure of a FQQSPR equation. Thus, this kind of relationship, if it exists, to some extent can be considered causal. A QQSPR permits us to compute the molecular properties of U-molecules, just employing non-empirical considerations and parameters or molecular descriptors of quantum mechanical origin. In this sense, these relationships can be considered universal, applicable to any molecular structure. However, due to the quantum mechanical nature of the QQSPR, one can obtain structure-properties relationships between QO's of any kind: nuclei, atoms, molecules. By QQSPR are here understood QSPR obtained by means of the application of canonical quantum theoretical methods to obtain expectation values of a Hermitian operator, which within QQSPR theory is described by functionals of the density function tags of a given QOS, acting as a basis set. QQSPR are more general than classical QSPR as they can be obtained for any QO, incorporating information difficult to take into account with classical molecular descriptors. The descriptors in the QQSPR framework are the DF tags of the QO.
- QS matrix:
-
Quantum Similarity Matrix. The matrix possessing positive definite elements, constructed by quantum similarity measures using the DF tags of the elements of a QOS. See: Quantum Similarity Matrices.
- Tagged set:
-
A set constructed as the Cartesian product of two separate sets. One of them, the object set is composed of well-defined elements of any nature, called objects. The other set in the tagged set is the tag set, made of some descriptors attached to every object, which is supposed to contain information about the objects; tags can be mathematical elements made by bit strings, vectors, matrices, functions…
- U-m:
-
Unknown (properties) molecule. A molecule which can be described as a QO, but lacks the property information associated to the molecules belonging to the Core Set. U-m properties act as the unknowns to be evaluated in the QQSPR theory developed here.
- U-molecule:
-
A U-m.
- U-m set:
-
A QOS containing U-m's as elements.
- Wave function:
-
A by-product of solving the Schrödinger equation. When studying stationary submicroscopic systems by means of classical quantum mechanics, the pair energy – wave function correspond to the eigenvalue – eigenfunction pairs of the Hermitian Hamiltonian operator constructed for the system, which substitutes the classical Hamilton function. The squared module of the stationary wave functions for each system state is customarily interpreted, since Born times, as the probability density function to find the system as a whole in some space infinitesimal element of volume.
Bibliography
Primary Literature
Carbó R, Arnau J, Leyda L (1980) How similar is a molecule to another? An electron density measure of similarity between two molecular structures. Int J Quantum Chem 17:1185–1189
Carbó R, Besalú E (1996) Mendeleev conjecture as a consequence of mendeleev postulates. Afinidad 53:77–79
Messiah A (1999) Quantum mechanics. Dover Publications, New York
Bohm D (1989) Quantum theory. Dover Publications, New York
Carbó-Dorca R, Robert D, Amat L, Girones X, Besalu E (2000) Molecular quantum similarity in QSAR and drug design. In: Lecture notes in chemistry, vol 73. Springer, Berlin
Carbó-Dorca R (1995) Molecular similarity and reactivity: From quantum chemical to phenomenological approaches. Kluwer Academic, Dordrecht
Carbó-Dorca R, Besalu E (1998) A general survey of molecular quantum similarity. J Mol Struct Theochem 451:11–23
Carbó-Dorca R (2000) European congress on computational methods in applied sciences and engineering. Barcelona, pp 1–31
Carbó-Dorca R, Besalu E (2000) Quantum theory of QSAR. Contributions Sci 1:399–422
Bultinck P, Girones X, Carbó-Dorca R (2005) Molecular quantum similarity: Theory and applications. In: Lipkowitz KB, Larter R, Cundari T (eds) Reviews in computational chemistry, vol 21. Wiley, Hoboken, pp 127–207
Carbó-Dorca R, Mezey PG (1998) Advances in molecular similarity. JAI Press, London
Carbó-Dorca R (2005) Mathematical elements of quantum electronic density functions. In: Advances in quantum chemistry, vol 49. Elsevier Academic, San Diego, pp 121–208
Carbó-Dorca R, Besalu E (2001) Extended Sobolev and Hilbert spaces and approximate stationary solutions for electronic systems within non-linear Schrödinger equation. J Math Chem 29:3–20
Carbó-Dorca R, Besalu E (2002) Fundamental quantum QSAR (QQSPR) equation: Extensions, nonlinear terms and generalizations within extended Hilbert–Sobolev spaces. Int J Quantum Chem 88:167–182
Johnson MA, Maggiora GM (1990) Concepts and applications of molecular similarity. Wiley, New York
Carbó R, Calabuig B, Besalú E, Martínez A (1992) Triple density molecular quantum similarity measures: A general connection between theoretical calculations and experimental results. Mol Eng 2:43–64
Carbo R, Calabuig B, Vera L, Besalu E (1994) Molecular quantum similarity: Theoretical framework, ordering principles, and visualization techniques. In: Advances in quantum chemistry, vol 25. Academic Press, San Diego, pp 253–313
Carbó-Dorca R (2001) Inward matrix products: Extensions and applications to quantum mechanical foundations of QSAR. J Mol Struct Theochem 537:41–54
Carbó-Dorca R, Amat L, Besalu E, Girones X, Robert D (2000) Quantum mechanical origin of QSAR: Theory and applications. J Mol Struct Theochem 504:181–228
Carbo R, Besalu E, Amat L, Fradera X (1995) Quantum molecular similarity measures (QMSM) as a natural way leading towards a theoretical foundation of quantitative structure-activity relationships (QSPR). J Math Chem 18:237–246
Carbó-Dorca R, Besalu E, Amat L, Fradera X (1996) Quantum molecular similarity measures: Concepts, definitions, and applications to quantitative structure-property relationships. In: Carbó-Dorca R, Mezey PG (eds) Advances in molecular similarity. JAI Press, London, pp 1–42
Carbó-Dorca R, Amat L, Besalú E, Gironés X, Robert D (2001) Quantum molecular similarity: Theory and applications to the evaluation of molecular properties, biological activities and toxicity. In: Carbó-Dorca R, Gironés X, Mezey PG (eds) Fundamentals of molecular similarity. Proc of 4th girona seminar on molecular similarity. Kluwer Academic/Plenum Publishers, New York, chap 12, pp. 187–320
Carbó-Dorca R (1997) Fuzzy sets and boolean tagged sets. J Math Chem 22:143–147
Carbó-Dorca R (1998) Tagged sets, convex sets and QS measures. J Math Chem 23:353–364
Carbó-Dorca R (1998) Fuzzy sets and boolean tagged sets; vector semispaces and convex sets; quantum similarity measures and ASA density functions; diagonal vector spaces and quantum chemistry. In: Carbó-Dorca R, Mezey PG (eds) Advances in molecular similarity. JAI Press, London, pp 43–72
Robert D, Girones X, Carbó-Dorca R (2000) Quantification of the influence of single point mutations on haloalkane dehalogenase activity: A molecular quantum similarity study. J Chem Inf Comput Sci 40:839–846
Robert D, Amat L, Carbó-Dorca R (2000) Quantum similarity QSAR: Study of inhibitors binding to trhombin, trypsin and factor Xa, including a comparision with CoMFA and CoMSIA methods. Int J Quantum Chem 80:265–282
Girones X, Gallegos A, Carbó-Dorca R (2000) Modelling antimalarial activity: Application of kinetic energy density quantum similarity measures as descriptors in QSAR. J Chem Inf Comput Sci 40:1400–1407
Robert D, Girones X, Carbó-Dorca R (2000) Molecular quantum similarity measures as descriptors for quantum QSAR. Polycycl Aromat Compd 19:51–71
Renners I, Ludwig LA, Grauel A, Benfenati E, Pellagatti S, Robert D, Carbó-Dorca R, Girones X (2000) Modeling toxicity with molecular descriptors and similarity measures via B-spline networks. IPMU2000 8th international conference on information processing and management of uncertainty in knowledge based systems, Madrid, Spain, pp 1021–1026
Renners I, Carbó-Dorca R, Grauel A, Ludwig LA, Robert D, Girones X (2000) Toxicity prediction by using genetically optimized B-spline networks based on molecular quantum similarity. 2nd international ICSC symposium on neural computation, Berlin, Germany
Gallegos A, Robert D, Girones X, Carbó-Dorca R (2001) Structure-toxicity relationships of polycyclic aromatic hydrocarbons using molecular quantum similarity. J Comput Aided Mol Design 15:67–80
Besalú E, Gallegos A, Carbó-Dorca R (2001) Topological quantum similarity indices and their use in QSAR: Application to several families of antimalarial compounds. Match-Commun Math Comput Chem 44:41–64
Gironés X, Carbó-Dorca R (2001) Sobre les relacions lineals d'energia lliure: Mesures de Semblança Molecular Quàntica sobre Funcions de Densitat Electrònica Modificades. Scientia Gerund 25:5–15
Ponec R, Gironés X, Carbó-Dorca R (2002) Molecular basis of LFER. On the nature of inductive effects in aliphatic series. J Chem Inf Comput Sci 42:564–570
Gironés X, Gallegos A, Carbó-Dorca R (2001) Antimalarial activity of synthetic 1,2,4-trioxabes and cyclid peroxy ketals, a quantum similarity study. J Comput-Aided Mol Design 15:1053–1063
Gironés X, Carbó-Dorca R (2002) Molecular quantum similarity-based QSARs for binding affinities of several steroid sets. J Chem Inf Comput Sci 42:1185–1193
Gallegos Saliner AG, Amat L, Carbó-Dorca R, Schultz TW, Cronin MTD (2003) Molecular quantum similarity analysis of estrogenic activity. J Chem Inf Comput Sci 43:1166–1176
Amat L, Carbó-Dorca R, Cooper DL, Allan NL, Ponec R (2003) Structure-property relationships andmomentum-space quantities:Hammett s-constants. Mol Phys 101:3159–3162
Gironés X, Carbó-Dorca R, Ponec R (2003) Molecular basis of LFER. Modeling of the electronic substituent effect using fragment quantum self-similarity measures. J Chem Inf Comput Sci 43:2033–2038
Nino A, Munoz-Caro C, Carbó-Dorca R, Gironés X (2003) Rational modelling of the voltage-dependent K+ channel inactivation by aminopyridines. Biophys Chem 104:417–427
Gallegos A, Carbó-Dorca R, Ponec R, Waisser K (2004) Similarity approach to QSAR. Application to antimycobacterial benzoxazines. Int J Pharm 269:51–60
Gironés X, Carbó-Dorca R (2004) Molecular similarity and quantitative structure-activity relationships. In: Bultinck P, De Winter H, Langenaecker W, Tollenaere JP (eds) Similarity and quantitative structure-activity relationships in computational medicinal chemistry for drug discovery. Marcel Dekker, New York, pp 365–385
Giralt F, Espinosa G, Arenas A, Ferre-Gine J, Amat L, Gironés X, Carbó-Dorca R, Cohen Y (2004) Estimation of infinite dilution activity coefficients of organic compounds in Water with neural classifiers. AIChE J 50:1315–1343
Carbó-Dorca R, Gironés X (2005) Foundation of quantum similarity measures and their relationship to QSPR: Density function structure, approximations and application examples. Int J Quantum Chem 101:8–20
Gironés X, Carbó-Dorca R (2006) Modelling toxicity using molecular quantum similarity measures. QSAR Comb Sci 25:579–589
Myers H (1990) Classical and modern regression with applications. Duxbury Press, California
Wold S, Sjöström M, Eriksson L (1994) Partial least squares projections to latent structures (PLS) in chemistry. In: von Ragué Schleyer P, Allinger NL, Clark T, Gasteiger J, Kollman PA, Schaefer HF III, Screiner PR (eds) Encyclopedia of computational chemistry. Wiley, Chichester, pp 2006–2021
Geladi P, Kowalski BR (1986) Partial least-squares regression: A tutorial. Analytica Chimica Acta 185:1–17
Wold S, Johansson E, Cocchi M (1993) PLS-partial least-squares projections to latent structures. In: Kubinyi H (ed) 3D QSAR in drug design. ESCOM, Science Publishers BV, Leiden, chap 5
Montgomery DC, Peck EA, Vining GG (1992) Introduction to linear regression analysis. Wiley, New York
Whitley DC, Ford MG, Livingstone DJ (2000) J Chem Inf Comput Sci 40:1160–1168
Wold S (1978) Cross-validatory estimation of the number of components in factor and principal component models. Technometrics 20:397–405
Jolliffe IT (1986) Principal component analysis. Springer, New York
Cattell RB (1966) The scree test for the number of factors. Multivar Behav Res 1:245–276
Wold S, Eriksson L (1995) Statistical validation of QSAR results. In: Van de Waterbeemd H (ed) Chemometric methods in molecular design. VCH, New York, sect IV, pp 309–318
Allen DM (1974) The relationship between variable selection and data augmentation and a method for prediction. Technometrics 16:125–127
Shao J (1993) Linear-Model selection by cross-validation. J Am Stat Assoc 88:486–494
Carbó-Dorca R (2000) Stochastic transformation of quantum similarity matrices and their use in quantum QSAR models. Int J Quantum Chem 79:163–177
Carbó-Dorca R (2007) About the prediction of molecular properties using the fundamental quantum QSPR (QQSPR) equation. SAR QSAR Environ Res 18:265–284
Carbó-Dorca R, Van Damme S (2007) Theoretical chemistry accounts: Theory, computation, and modeling. Theoretica Chimica Acta 118:673–679
Carbó-Dorca R, Van Damme S (2007) Riemann spaces, molecular density function semispaces, quantum similarity measures and quantum quantitative structure-properties relationships (QQSPR). Afinidad 64:147–153
Hansch C, Fujita T (1962) J Am Chem Soc 86:10
Oliva JM, Carbó-Dorca R, Mestres J (1996) Conformational analysis from the point of view of quantum molecular similarity. In: Carbó-Dorca R, Mezey PG (eds) Advances in molecular similarity. JAI PRESS, London, pp 135–165
Carbó-Dorca R (2000) Quantum quantitative structure-activity relationships (QQSAR): A comprehensive discussion based on inward matrix products, employed as a tool to find approximate solutions of strictly positive linear systems and providings QSAR-quantum similarity measures connection. In: Proc of the european congress on computational methods in applied sciences and engineering (ECCOMAS 2000), Barcelona, chap 12
Sen K, Carbó-Dorca R (2000) Inward matrix products, generalised density functions and Rayleigh–Schrödinger perturbation Theory. J Mol Struct Theochem 501:173–176
Carbó-Dorca R (2001) Inward matrix product algebra and calculus as tools to construct space-time frames of arbitrary dimensions. J Math Chem 30:227–245
Carbó-Dorca R (2003) Applications of inward matrix products and matrix wave functions to Hückel MO theory, Slater extended wave functions, spin extended functions and Hartree method. Int J Quantum Chem 91:607–617
Carbó-Dorca R (2003) About some questions relative to the arbitrariness of signs: Their possible consequences in matrix signatures definition and quantum chemical applications. J Math Chem 33:227–244
Vinogradov IM (1989) Encyclopaedia of mathematics, vol 4. Kluwer Academic, Dordrecht, p 351
Lahey Computer Systems (1998) LF 95 Language Reference. Lahey Computer Systems, Incline Village http://www.lahey.com
Carbó-Dorca R (2002) Shell partition and metric semispaces: Minkowski norms, root scalar products, distances and cosines of arbitrary order. J Math Chem 32:201–223
Bultinck P, Carbó-Dorca R (2004) A mathematical discussion on density and shape functions, vector semispaces and related questions. J Math Chem 36:191–200
Vinogradov IM (1992) Encyclopaedia of mathematics, vol 8. Kluwer Academic, Dordrecht, p 249
Constans P, Amat LL, Fradera X, Carbó-Dorca R (1996) Quantum molecular similarity measures (QMSM) and the atomic shell approximation (ASA). In: Carbó-Dorca R, Mezey PG (eds) Advances in molecular similarity, vol 1. JAI Press, London, chap 8, pp 187–211
Gironés X, Amat L, Carbó-Dorca R (1998) A comparative study of isodensity surfaces using ab initio and ASA density functions. J Mol Graph Model 16:190–196
Constants P, Carbó R (1995) Atomic shell approximation: Electron density fitting algorithm restricting coefficients to positive values. J Chem Inf Comput Sci 35:1046–1053
Amat LL, Carbó-Dorca R (1997) Quantum similarity measures under atomic shell approximation: First order density fitting using elementaryJacobi rotations. J Comput Chem 18:2023–2039
Amat LL, Carbó-Dorca R (1999) Fitted electronic density functions from H to Rn for use in quantum similarity measures: Cis-diamminedichloroplatinum(II) complex as an application example. J Comput Chem 20:911–920
Amat LL, Carbó-Dorca R (2000) Molecular electronic density fitting using elementary Jacoby rotations under atomic shell approximation (ASA). J Chem Inf Comput Sci 40:1188–1198
Gironés X, Carbó-Dorca R, Mezey PG (2001) Application of promolecular ASA densities to graphical representation of density functions of macromolecular systems. J Mol Graph Model 19:343–348
Mestres J, Solà M, Besalú E, Duran M, Carbó R (1995) Electron density approximations for the fast evaluation of quantum molecular similarity measures. In: Carbó R (ed) Molecular similarity and reactivity: From quantum chemical to phenomenological approaches. Kluwer Academic, Dordrecht, pp 77–85
Mestres J, Solà M, Duran M, Carbó R (1995) General suggestions and applications of quantum molecular similarity measures from ab initio fitted electron densities. In: Carbó R (ed) Molecular similarity and reactivity: From quantum chemical to phenomenological approaches. Kluwer Academic, Dordrecht, pp 89–111
Carbó R, Calabuig B (1990) Molecular similarity and quantum chemistry. In: Johnson MA, Maggiora GM (eds) Concepts and applications of molecular similarity. Wiley, New York, chapt 6, p 147
Carbó R, Besalú E, Amat LL, Fradera X (1996) On Quantum Molecular Similarity Measures (QMSM) and Indices (QMSI). J Math Chem 19:47–56
Robert D, Carbó-Dorca R (2000) General trends in atomic and nuclear quantum similarity measures. Int J Quantum Chem 77:685–692
Besalú E, Carbó-Dorca R, Karwowski J (2001) Generalized one-electron spin functions and Self-Similarity Measures. J Math Chem 29:41–45
Amat L, Besalú E, Carbó-Dorca R (2001) Identification of active molecular sites using quantum-self similarity measures. J Chem Inf Comput Sci 41:978–991
Besalú E, Gironés X, Amat L, Carbó-Dorca R (2002) Molecular quantum similarity and the fundaments of QSAR. Acc Chem Res 35:289–295
Amat L, Carbó-Dorca R, Cooper DL, Allan NL (2003) Classification of reaction pathways via momentum-space and quantum molecularsimilarity measures. Chem Phys Lett 367:207–213
Bultinck P, Carbó-Dorca R (2003) Molecular quantum similarity matrix based clustering of molecules using dendrograms. J Chem Inf Comput Sci 43:170–177
Ponec R, Bultinck P, Van Damme S, Carbó-Dorca R, Tantillo DJ (2005) Geometric and electronic similarities between transition structures for electrocyclizations and sigmatropic hydrogen shifts. Theor Chem Acc 113:205–211
Bultinck P, Carbó-Dorca R (2005) Molecular quantum similarity using conceptual DFT descriptors. J Chem Sci 117:425–435
Carbó-Dorca R (2008) A quantum similarity matrix (QSM) Aufbau procedure. J Math Chem 44:228–234
Berberian SK (1961) Introduction to Hilbert space. Oxford University Press, New York
Carbó R, Besalú E (1992) Many center AO integral evaluation using cartesian exponential type orbitals (CETO's). Adv Quantum Chem 24:115–237
von Neumann J (1955) Mathematical foundations of quantum mechanics. Princeton University Press, Princeton
Born M (1945) Atomic physics. Blackie and Son, London
Dirac PAM (1983) The principles of quantum mechanics. Clarendon Press, Oxford
Carbó-Dorca R (2000) Quantum QSAR and the eignsystems of stochastic quantum similarity matrices. J Math Chem 27:357–376
Gironés X, Amat L, Carbó-Dorca R (2000) Use of electron-electron repulsion energy as a molecular description in QSAR or QSPR studies. J Comput Aided Mol Design 14:477–485
Kubinyi H (1993) 3D QSAR in drug design. Theory, methods and applications. ESCOM Science Publishers, Leiden
Martin YC (1978) Quantitative drug design. Medicinal research series, vol 8. Marcel Dekker, New York
Lien EJ (1987) SAR, side effects and drug design. Medicinal research series, vol 11. Marcel Dekker, New York
Martin YC, Kutter E, Austel V (1989) Modern drug research. Medicinal research series, vol 12. Marcel Dekker, New York
Wermuth CG (1993) Trends in QSAR molecular modelling 92. Escom, Leiden
Kubinyi H (1998) Quantitative structure-activity relationships in drug design. In: Schleyer PVR, Allinger NL, Clark T, Gasteiger J, Kollman PA, Schaefer HF III, Schreiner PR (eds) Encyclopedia of computational chemistry, vol 4. Wiley, Chichester, pp 2309–2319
Charton M (1996) Advances in quantitative structure-property relationships, vol 1. JAI Press, London
Van de Waterbeemd H (1996) Structure-property correlations in drug research. Academic Press, San Diego
Jurs PC (1998) Quantitative structure-property relationships (QSPR). In: Schleyer PVR, Allinger NL, Clark T, Gasteiger J, Kollman PA, Schaefer HF III, Schreiner PR (eds) Encyclopedia of computational chemistry, vol 4. Wiley, Chichester, pp 2320–2330
Boethling RS, Mackay D (2000) Handbook of property estimation methods for chemicals. Environmental and health sciences. Lewis Publishers, London
Neter J, Wasserman W, Kutner MH (1990) Applied linear statistical models. RD Irwin, Boston
Wagner M, Sadowski J, Gasteiger J (1995) Autocorrelation of molecular surface properties for modeling corticosteroid binding globulin and cytosolic Ah receptor activity by neural networks. J Am Chem Soc 117:7769–7775
Cramer RD, Patterson DE, Bunce JD (1988) Comparative molecular field analysis (CoMFA). 1. Effect of shape on binding of steroids tocarrier proteins. J Am Chem Soc 110:5959–5967
Parretti MF, Kroemer RT, Rothman JH, Richards WG (1997) Alignment of molecules by Monte Carlo optimisation and molecular similarity indices. J Comput Chem 18:1344–1353
So SS, Karplus M (1997) Three-dimensional quantitative structure-activity relationships from molecular similarity matrices and genetic neural networks. 1. Method and validations. J Med Chem 40:4347–4359
Jain AN, Koile K, Chapman D (1994) Compass: Predicting biological activities from molecular surface properties. Performance comparisons on a steroid benchmark. J Med Chem 37:2315–2327
Bravi G, Gancia E, Mascagni P, Pegna M, Todeschini R, Zaliani A (1997) MS-WHIM, new 3D theoretical descriptors derived from molecular surface properties: A comparative 3D QSAR study in a series of steroids. J Comput Aided Mol Design 11:79–92
Norinder U (1990) Experimental design based 3D-QSAR analysis of steroid-protein interactions: Application to human CBG complexes. J Comput Aided Mol Design 4:381–389
Norinder U (1991) 3D-QSAR analysis of steroid/protein interactions: The use of difference maps. J Comput Aided Mol Design 5:419–426
Rum G, Herndon WC (1991) Molecular similarity concepts. 5. Analysis of steroid-protein binding constants. J Am Chem Soc 113:9055–9060
Simon Z, Bohl M (1992) Structure-activity relations in gestagenic steroids by the MTD method. The case of hard molecules and soft receptors. Quant Struct Activity Relatsh 11:23–28
Waszkowycz B, Clark DE, Frenkel D, Li J, Murray CW, Robson B, Westhead DR (1994) PRO_LIGAND: An approach to de novo molecular design. 2. Design of novel molecules from molecular field analysis (MFA) models and pharmacophores. J Med Chem 37:3994–4002
Dunn WJ, Wold S, Edlund U, Hellberg S (1984) Multivariate structure-activity relationships between data from a battery of biological tests and an ensemble of structure descriptors: the PLS method. Quant Struct Activity Relatsh 3:131–137
Good AC, So SS, Richards WG (1993) Structure-activity relationships from molecular similarity matrices. J Med Chem 36:433–438
Oprea TI, Ciubotariu D, Sulea TI, Simon Z (1993) Comparison of the minimal steric difference (MTD) and comparative molecular field analysis (CoMFA) methods for analysis of binding of steroids to carrier proteins. Quant Struct Activity Relatsh 12:21–26
Klebe G, Abraham U, Mietzner T (1994) Molecular similarity indices in a comparative analysis (CoMSIA) of drug molecules to correlate and predict their biological activity. J Med Chem 37:4130–4146
Hahn M, Rogers D (1995) Receptor Surface Models. 2. Application to quantitative structure-activity relationships studies. J Med Chem 38:2091–2102
Silverman BD, Platt DE (1996) Comparative molecular moment analysis (CoMMA): 3D-QSAR without molecular superposition. J Med Chem 39:2129–2140
Kellogg GE, Kier LB, Gaillard P, Hall LH (1996) E-state fields: Applications to 3D QSAR. J Comput Aided MolDesign 10:513–520
Anzali S, Barnickel G, Krug M, Sadowski J, Wagener M, Gasteiger J, Polanski J (1996) The comparison of geometric and electronic properties of molecular surfaces by neural networks: Application to the analysis of corticosteroid-binding globulin activity of steroids. J Comput Aided Mol Design 10:521–534
Norinder U (1996) 3D-QSAR Investigation of the tripos benchmark steroids and some protein-tyrosene kinase inhibitors of styrene type using the TDQ approach. J Chemom 10:533–545
Schnitker J, Gopalaswamy R, Crippen GM (1997) Objective models for steroid binding sites of human globulins. J Comput Aided Mol Design 11:93–110
Turner DB, Willett P, Ferguson AM, Heritage T (1997) Evaluation of a novel infrared range vibration-based descriptor (EVA) for QSAR studies. 1. General application. J Comput Aided Mol Design 11:409–422
Cramer RD III, Depriest SA, Patterson DE, Hecht P (1993) The developing practice of comparative molecular field analysis. In: Kubinyi H (ed) 3D QSAR in drug design. ESCOM, Leiden, part III, pp 443–485
Good AC (1995) 3D molecular similarity indices and their application in QSAR studies. In: Dean PM (ed) Molecular similarity in drug design. Blackie Academic & Professional (Capman & Hall), London, chap 2
Ortiz AR, Pisabarro MT, Gago F, Wade RC (1995) Prediction of drug binding affinities by comparative binding energy analysis: Application to human synovial fluid phospholipase A2 inhibitiors. In: Sanz F, Giraldo J, Manaut F (eds) QSAR and molecular modelling: Concepts, computational tools and biological applications. Prous Pub, Barcelona, sect VII, pp 439–443
Oprea TI, Head RD, Marshall GR (1995) The basis of cross-reactivity for a series of steroids binding to monoclonal antibody (DB3) against progesterone. A molecular modeling and QSAR study. In: Sanz F, Giraldo J, Manaut F (eds) QSAR and molecular modelling: Concepts, computational tools and biological applications. Prous Pub, Barcelona, sect VII, pp 461–462
Kimura T, Hasegawa K, Funatsu K (1998) GA strategy for variable selection in QSAR studies: GA-based region selection for CoMFA modeling. Chem Inf Comput Sci 38:276–282
Mabilia M, Belvisi L, Bravi G, Catalano G, Scolastico C (1995) A PCA/PLS analysis on nonpeptide angiotensin II receptor antagonists. In: Sanz F, Giraldo J, Manaut F (eds) QSAR and molecular modelling: Concepts, computational tools and biological applications. Prous Pub,Barcelona, sect VII, pp 456–460
Sulea T, Oprea T, Muresan S, Ling Chan S (1997) A different method for steric field evaluation in CoMFA improves model robustness. Chem Inf Comput Sci 37:1162–1170
Palomer A, Giolitti A, García ML, Fos E, Cabré F, Mauleón D, Carganico G (1995) Molecular modeling and CoMFA investigations on LTD4 receptor antagonists. In: Sanz F, Giraldo J, Manaut F (eds) QSAR and molecular modelling: Concepts, computational tools and biological applications. Prous Pub, Barcelona, sect VII, pp 444–450
Tetko IV, Villa AEP, Livingstone DJ (1996) Neural network studies. 2. variable selection. Chem Inf ComputSci 36:794–803
Klebe G, Abraham U, Mietzner T (1994) Molecular similarity indices in comparative analysis (CoMSIA) of drug molecules to correlate and predict their biological activity. J Med Chem 37:4130–4146
Richon AB, Young SS (2008) An introduction to QSAR methodology. A web paper of the network science corporation: http://www.netsci.org/Science/Compchem/feature19.html. Accessed 27 Oct 2008
So SS, van Helden SP, van Geerestein VJ, Karplus M (2000) Quantitative structure-activity relationship studies of progesterone receptor binding steroids. Chem Inf Comput Sci 40:762–772
Gantmacher F R (1966) Théorie des matrices, vol 2. Dunod, Paris
Jacobi CGJ (1846) J Reine Angew Math 30:51–94
Carbó-Dorca R, Besalú E, Gironés X (2000) Extended density functions. Adv Quantum Chem 38:3–63
Eyring H, Walter J, Kimball GE (1944) Quantum chemistry. Wiley, New York
Hoffmann K (1988) Banach spaces and analytic functions. Dover Publications, New York
Akhiezer NI, Glazman IM (1993) Theory of linear operators in Hilbert space. Dover Publications, New York
Landau LD, Lifshitz EM (1967) Mecánica Cuántica No-Relativista. Editorial Reverté, Barcelona
Vinogradov IM (1992) Encyclopaedia of mathematics, vol 8. Reidel-Kluwer Academic, Dordrecht, pp 379
Stewart J (1991) Advanced General Relativity. Cambridge University Press, Cambridge
Bach A, Amat L, Besalú E, Carbó-Dorca R, Ponec R (2000) Quantum chemistry, Sobolev spaces and SCF. J Math Chem 28:59–70
Carbó-Dorca R (2002) Density functions and generating wave functions. In: Sen K (ed) Reviews in modern quantum chemistry, vol I, 15, pp 401–412
Edwards CH Jr (1994) Advanced calculus of several variables. Dover Publications, New York
Carbó R, Besalú E (1996) Applications of nested summation symbols to quantum chemistry: Formalism and programming techniques. In: Ellinger Y, Defranceschi M (eds) Strategies and applications in quantum chemistry. Kluwer Academic, Dordrecht, pp 229
Carbó R, Besalú E (1995) Definition and quantum chemical applications of nested summation symbols and logical functions: Pedagogical artificial intelligence devices for formulae writing, sequential programming and automatic parallel implementation. J Math Chem 18:37–72
BreitTG; Phys. Revs. 34 (1929) 553–573; Ibid. 36 (1930) 383–397; Ibid. 39 (1932) 616–624
Bethe HA, Salpeter EE (1957) Quantum mechanics of one- and two-electron systems. Springer, Berlin
Moss RE (1973) Advanced molecular quantum mechanics. Chapman and Hall, London
Carbó-Dorca R, Karwowski J (2001) Theoretical and computational aspects of extended wave functions. Int J Quantum Chem 84:331–337
McWeeny R, Sutcliffe BT (1969) Methods of molecular quantum mechanics. Academic Press, London
Huzinaga S (1991) Theochem 234:51–73
Huzinaga S, Katsuki S, Matsuoka O (1992) J Math Chem 10:25–39
Bonifacic V, Huzinaga S (1974) J Chem Phys 60:2779–2786
Carbó R, Calabuig B (1992) Molecular quantum similarity measures and n-dimensional representation of quantum objectsI. theoretical foundations. Int J Quantum Chem 42:1681–1709
Carbó-Dorca R, Besalú E (1996) Extending molecular similarity to electronic energy surfaces: Boltzmann similarity measures and indices. J Math Chem 20:247–261
Bultinck P, De Winter H, Langenaecker W, Tollenaere JP (2004) Similarity and quantitative structure-activity relationships in computational medicinal chemistry for drug discovery. Marcel Dekker, New York
Vinogradov IM (1988) Encyclopaedia of mathematics, vol 1. Reidel-Kluwer Academic, Dordrecht, p 334
Korn GA, Korn TM (1967) Manual of mathematics. Mc Graw-Hill, New York
Stewart J (1999) Cálculo. International Thompson Editores, México
Carbó R, Besalú E (1993) J Math Chem 13:331–342
Carbó R, Besalú E (1994) Comput Chem 18:117–126
Carbó-Dorca R (2000) A discussion on an apparent MO theory paradox. J Math Chem 27:35–41
Fraga S, García de la Vega JM, Fraga ES (1999) The Schrödinger and Riccati equations. Lecture notes in chemistry, vol 70. Springer, Berlin
Meyer CD (2000) Matrix analysis and applied linear algebra. SIAM, Philadelphia
Amat LL, Carbó-Dorca R, Ponec R (1999) Simple linear QSAR models based on quantum similarity measures. J Med Chem 42:5169–5170
Besalú E, Carbó R, Mestres J, Solà M (1995) In: Sen K (ed) Molecular similarity I. Top Curr Chem 173:31
Hansch C, Fujita T (1964) J Am Chem Soc 86:5175
Amat LL, Carbó-Dorca R, Ponec R (1998) Molecular quantum similarity studies as an alternative to log P values in QSAR studies. J Comput Chem 14:1575–1583
Carbó R, Besalú E (1994) Comput Chem 18:117
Books and Reviews
Al-Fahemi J, Cooper DL, Allan NL (2005) The quantitative use of momentum-space descriptors. Chem Phys Lett 416:376–380
Al-Fahemi J, Cooper DL, Allan NL (2005) The use of momentum-space descriptors for predicting octanol-water partition coefficients. J Mol Struct Theochem 727:57–61
Allan NL, Cooper DL (1995) Topics Curr Chem 173:85–111
Amat LL, Carbó-Dorca R (2002) Use of promolecular ASA density functions as a general algorithm to obtain starting MO in SCF calculations. Int J Quantum Chem 87:59–67
Amat LL, Besalú E, Carbó R, Fradera X (1995) Practical applications of quantum molecular similarity measures (QMSM): Programs and examples. Scientia Gerund 21:17–34
Amat LL, Constans P, Carbó R (1996) Descripció d'un algorisme d'optimització global de les Mesures de Semblança Quàntica Molecular. Scientia Gerund 22:109–121
Amat LL, Pradera X, Carbó R (1996) Sobre els mapes de Semblança Quàntica Molecular. Scientia Gerund 22:97–107
Amat LL, Robert D, Besalú E, Carbó–Dorca R (1998) Molecular quantum similarity measures tuned QSAR: An antitumoral family validation study. J Chem Inf Comput Sci 38:624–631
Amovilli C, McWeeny R (1991) J Mol Struct Theochem 227:1–9
Antolín J, Angulo JC (2008) Quantum similarity indices for atomic ionization processes. Eur Phys J D 46:21–26
Atkins PW, Friedman RS (1997) Molecular quantum mechanics. Oxford University Press, Oxford
Bach A, Carbó-Dorca R (1999) Aplicació de la Semblança Molecular Quàntica en la reducció de l'espai configuracional per a l'estat fonamental i primers exitats de l'àtom d'heli. Scientia Gerund 24:183–196
Bader RFW (1990) Atoms in molecules. Clarendon Press, Oxford
Bell JS (1993) Speakable and unspeakable in quantum mechanics. Cambridge University Press, Cambridge
Benigni R, Cotta-Ramusino M, Giorgi F, Gallo G (1995) J Med Chem 38:629–635
Benigni R, Cotta-Rasmusino M, Gallo G, Giorgi F, Giuliani A, Vari MR (2000) J Med Chem 43:3699–3707
Besalú E (2001) Fast computation of cross-validated properties in full linear leave-many-out procedures. J MathChem 29:191–204
Besalú E, Carbó R (1995) Quantum similarity topological indices: Definition, analysis and comparison with classical molecular topological indices. Scientia Gerund 21:145–152
Besalú E, Amat L, Fradera X, Carbó R (1995) An application of the molecular quantum similarity: Ordering of some properties of the hexanes. In: QSAR and molecular modelling: Concepts, computational tools and biological applications. Proc of the 10th european symposium on structure-activity relationships. Prous Science, Barcelona, pp 396–399
Besalú E, Carbó R, Mestres J, Solà M (1995) Foundations and recent developments of quantum molecular similarity. In: Topics in current chemistry: Molecular similarity I, vol 173. Springer, Berlin, pp 31–62
Besalú E, Carbó R, Duran M, Mestres J (1995) MESSEM: A density-based quantum molecular similarity system of programs. In: Clementi E (ed) Methods and techniques in computational chemistry METECC-95. STEF, Cagliari, pp 491–508
Bethe HA, Jackiw R (1986) Intermediate quantum mechanics. Benjamin, Menlo Park
Bonaccorsi R, Scrocco E, Tomasi J (1970) J Chem Phys 52:5270–5284
Boon G, Van Alsenoy C, De Proft F, Bultinck P, Geerlings P (2005) Molecular quantum similarity of enantiomers of amino acids: a case study. J Mol Struct Theochem 727:49–56
Borgoo A, Godefroid M, Sen KD, De Proft F, Geerlings P (2004) Quantum similarity of atoms: A numerical Hartree–Fock and information theory approach. Chem Phys Lett 399:363–367
Borgoo A, Torrent-Sucarrat M, De Proft F, Geerlings P (2007) Quantum similarity study of atoms: A bridge between hardness and similarity indices. J Chem Phys 126:234104
Born M (1944) Atomic physics. Blackie & Son, London
Botella V, Pacios LF (1998) Analytic atomic electron densities in molecular se If-similarity measures and electrostatic potentials. J Mol Struct Theochem 426:75–85
Bultinck P, Carbó-Dorca R, Van Alsenoy C (2003) Quality of approximate electron densities and internal consistency of molecular alignment algorithms in molecular quantum similarity. J Chem Inf Comput Sci 43:1208–1217
Bultinck P, Rafat M, Ponec R, Van Gheluwe B, Carbó-Dorca R, Popelier P (2006) Coulomb and overlap self-similarities: A comparative selectivity analysis of structure- electron delocalization and aromaticity in linear polyacenes: Atoms in molecules multicenter delocalization ind. J Phys Chem A 110:7642–7648
Bultinck P, Ponec R, Gallegos A, Fias S, Van Damme S, Carbó-Dorca R (2006) Generalized Polansky index as an aromaticity measure in polycyclic aromatic hydrocarbons. Croatica Chemica Acta 79: 363–371
Bunge M (1979) Causality in modern science. Dover, New York
BurtC, Richards WG, Huxley PH (1990) J Comput Chem 11:1139–1146
Carbó R (1995) Molecular similarity and reactivity: From quantum chemical to phenomenological approaches. Understanding chemical reactivity, vol 14. Kluwer Academic, Dordrecht
CarbóR, Besalú E (1995) Theoretical Foundations of Quantum Molecular Similarity. In: Carbó R (ed) Molecular similarity and reactivity: From quantum chemistry to phenomenological approaches. Kluwer Academic, Dordrecht, pp 3–30
Carbó R, Calabuig B (1989) Molsimil-88: Molecular similarity calculations using a CNDO approximation. Comput Phys Commun 55:117
Carbó R, Calabuig B (1989) Sobre las Medidas de Semejanza Molecular: Una conexión entre Química Quántica e Inteligencia Artificial. Una contribucion a los Anales de la VI Escuela Latinoamericana de Química Teórica, Brasil, vol 1, pp 134
Carbó R, Calabuig B (1992) Molecular quantum similarity measures and n-dimensional representation of quantum objects II. Practical applications. Int J Quantum Chem 42:1695
Carbó R, Calabuig B (1992) Quantum molecular similarity measures and the n-dimensional representation of a molecular set:Phenyldimethylthiazines. J Mol Struct Theochem 254:517–531
Carbó R, Calabuig B (1992) Quantum similarity measures, molecular cloud description and structure-properties relationships. J Chem Inf Comput Sci 32:600–606
Carbó R, Calabuig B (1992) Quantum similarity: Definitions, computational details and applications. In: Fraga S (ed) Computational chemistry: Structure, interactions and reactivity. Elsevier, Amsterdam, pp 300–324
Carbó R, Domingo LL (1987) LCAO MO similarity measures and taxonomy. Int J Quantum Chem 32:517–545
Carbó R, Riera JM (1978) A general SCF theory. Lecture notes in chemistry, vol 5. Springer, Berlin
Carbó R, Martín M, Pons V (1977) Application of quantum mechanical parameters in quantitative structure-activity relationships, Afinidad 34:348–353
Carbó R, Suñé E, Lapeña F, Pérez J (1986) Electrostatic potential comparison and molecular metric spaces. J Biol Phys 14:21
Carbó R, Lapeña F, Suñé E (1986) Similarity measures on electrostatic molecular potentials. Afinidad 43:483
Carbó R, Calabuig B, Martínez A (1991) Semblança molecular: Representació n-dimensional d'un conjunt de molècules. Scientia Gerund 17:133
Carbó R, Molino L, Calabuig B, Besalú E, Martínez A (1992) Medidas de Semejanza Cuántica: Conceptos Clásicos y Nuevas Estrategias. Folia Chimica Theoretica Latina 21–45
Carbó-Dorca R (1998) On the statistical interpretation of density functions: ASA, convex sets, discrete quantum chemical molecular representations, diagonal vector spaces and related problems. J Math Chem 23:365–375
Carbó-Dorca R (2000) Quantum QSAR and the eigensystems of stochastic quantum similarity matrices. J Math Chem 27:357–376
Carbó-Dorca R (2004) Heisenberg's relations in discrete n-dimensional parameterized metric vector spaces. J Math Chem 36:41–54
Carbó-Dorca R (2004) Infinite-dimensional time vectors as background building blocks of a space-time frame structure. J Math Chem 36:75–81
Carbó-Dorca R (2005) Molecular nuclear fields: A naïve perspective. J Math Chem 38:671–676
Carbó-Dorca R (2004) Non-linear terms & variational approach in quantum QSPR. J Math Chem 36:241–260
Carbó-Dorca R (2006) Descriptors and probability distributions in MO theory: Weighted Mulliken matrices and molecular quantum similarity measures. J Math Chem 39:551–591
Carbó-Dorca R (2008) Mathematical aspects of the LCAO MO first order density function (1): Atomic partition, metric structure and practical applications. J Math Chem 43:1076–1101
Carbó-Dorca R (2008) Mathematical aspects of the LCAO MO first order density function (2): Relationships between density functions. J Math Chem 43:1102–1118
Carbó-Dorca R, Besalú E (2006) Generation of molecular fields, quantum similarity measures and related questions. J Math Chem 39:495–509
Carbó-Dorca R, Bultinck P (2004) A general procedure to obtain quantum mechanical charge and bond order molecular parameters. J Math Chem 36(3):201–210
Carbó-Dorca R, Bultinck P (2004) Quantum mechanical basis for Mulliken population analysis. J Math Chem 36:231–239
Carbó-Dorca R, Bultinck P (2008) Mathematical aspects of the LCAO MO first order density function (3): A general localization procedure. J Math Chem 43:1069–1075
Chaves J, Barroso J, Bultinck P, Carbó-Dorca R (2006) Toward an alternative hardness kernel matrix structure in theelectronegativity equalization method (EEM). J Chem InfModel 46:1657–1665
CioslowskiJ, Challacombe M (1991) Int J Quantum Chem S25:81–93
Cioslowski J, Fleishmann ED (1991) J Am Chem Soc 113:64–67
Cioslowski J, Mixon ST (1992) Can J Chem 70:443–449
CioslowskiJ, Nanayakkara A (1993) J Am Chem Soc 115:11213–11215
Cioslowski J, Stefanov BB, Constans P (1996) J Comput Chem 17:1352–1358
Constans P, Amat L, Carbó-Dorca R (1997) Towards a global maximization of the molecular similarity function: The superposition of two molecules. J Comput Chem 18:826–846
Cooper DL, Allan NL (1989) J Comput Aided Mol Design 3:253–259
Cooper DL, Allan NL (1992) J Am Chem Soc 114:4773–4776
Cooper DL, Mort KA, Allan NL, Kinchington D, McGuidan CH (1993) J Am Chem Soc 115:12615–12616
Davidson ER (1976) Reduced density matrices in quantum chemistry. Academic Press, New York
Davydov AS (1965) Quantum mechanics. Pergamon Press, New York
Dean PM (1995) Molecular similarity in drug design. Blackie Academic & Professional, London
Dedekind R (1963) Essays on the theory of numbers. Dover, New York
Dunn JF, Nisula BC, Rodbard D (1981) J Clin Endocrinol Metab 53:58–68
Eyring H, Walker J, Kimball GE (1948) Quantum chemistry. Wiley, New York
Ferro N, Gallegos A, Bultinck P, Jacobsen HJ, Carbó-Dorca R, Reinard T (2006) Coulomb and overlap self-similarities: A comparative selectivity analysis of structure-function relationships for Auxin-like molecules. J Chem Inf Model 46:1751–1762
Ferro N, Bultinck P, Gallegos A, Jacobsen HJ, Carbó-Dorca R, Reinard T (2007) Unrevealed structural requrements for auxin-like molecules by theoretical and experimental evidences. Phytochemistry 68:237–250
Fradera X, Amat LL, Besalú E, Carbó-Dorca R (1997) Application of molecular quantum similarity to QSAR. Quant Struct Activity Relatsh 16:25–32
Fratev F, Monev V, Mehlhorn A, Polansky OE (1979) J Mol Struct 56:255–266
Fratev F, Polansky OE, Mehlhorn A, Monev V (1979) J Mol Struct 56:245–253
Gallegos A (2006) Mini-review on chemical similarity and prediction of toxicity. Curr Comput Aided Drug Des 2:105–122
Gallegos A, Girones X (2005) Theoretical Background and QSPR Application. J Chem Inf Model 45:321–326
Gallegos A, Girones X (2005) Topological quantum similarity measures: Applications in QSAR. J Mol Struct Theochem 727:97–106
Gallegos A, Patlewicz G, Worth AP (2005) The use of similarity measures in defining the applicability domain of skin sensitisation SARs. Altex-Alternativen zu Tierexperimenten 22:272
Gallegos A, Netzeva TI, Worth AP (2006) Prediction of estrogenicity: Validation of a classification model. SAR QSAR Environ Res 17:195–223
Gallegos A, Tsakovska I, Pavan M, Patlewicz G, Worth A (2007) Evaluation of SARs for the prediction of skin irritation/corrosion potential-structural inclusion rules in the BfR decision support system. SAR QSAR Environ Res 18:331–342
Gallegos A, Patlewicz G, Worth AP (2008) A review of (Q)SAR models for skin and eye irritation and corrosion. QSAR and Combinatorial Sciences 27:49–59
Geerlings P, De Proft F (2002) Chemical reactivity as described by quantum chemical methods. Int J Mol Sci 3:276–309
Geerlings P, Boon G, Van Alsenoy C, De Proft F (2005) Density functional theory and quantum similarity. Int J Quantum Chem 101:722–732
Gironés X, Amat L, Carbó-Dorca R (1999) Using molecular quantum similarity measures as descriptors in quantitative structure-toxicity relationships. SAR QSAR Environ Res 10:545–556
Gironés X, Robert R, Carbó-Dorca R (2001) TGSA: A molecular superposition program based on topo-geometrical considerations. J Comput Chem 22:255–263
Gironés X, Amat L, Carbó-Dorca R (2002) Modeling large macromolecular structures using promolecular densities. J Chem Inf Comput Sci 42:847–852
Gironés X, Carbó-Dorca R (2004) TGSA-Flex: Extending the capabilities of the topo-geometrical superposition algorithm to handle rotary bonds. J Comput Chem 25:153–159
Goldstein S (1988) Physics Today, March, 42 and April, 38
Good AC (1992) J Mol Graph 10:144–151
Good AC, Richards WG (1993) J Chem Inf Comput Sci 33:112–116
Good AC, Hodgkin EE, Richards WG (1992) J Chem Inf Comput Sci 32:188–191
Greiner W (1997) Relativistic quantum mechanics. Springer, Berlin
Gruber PM, Wills JM (1993) Handbook of convex geometry. North-Holland, Amsterdam
HoM, Smith VH, Weaver DF, Gatti C (1998) J Chem Phys 108:5469–5475
Hodgkin EE, Richards WG (1987) Int J Quantum Chem 14 :105–110
HuzinagaS, Klobukowski M (1998) J Mol Struct Theochem 167:1–209
Jeffrey A (1995) Handbook of mathematical formulas and integrals. Academic Press, New York
Klein J, Babic D (1997) Chem Inf Comput Sci 37:656
Lee CH, Smithline SH (1994) J Phys Chem 98:1135–1138
Leherte L (2006) Similarity measures based on Gaussian-type promolecular electron density models: Alignment of small rigid molecules. J Comput Chem 27:1800–1816
Lobato M, Besalú E, Carbó R (1996) Relacions Estructura-Propietat per un conjunt d'hidrucarburs a partir de nous descriptors tridimensionals derivats de la Semblança Molecular. Scientia Gerund 22:79–86
Lobato M, Amat L, Besalú E, Carbó-Dorca R (1997) Structure-activity relationships of a steroid family using quantum similarity measures and topological quantum similarity indices. Quant Struct Activity Relatsh 16:465–472
Lobato M, Amat L, Besalú E, Carbó-Dorca R (1998) Estudi QSAR d'una família de quinolones utilitzant indexs de semblança i indexs topològics de semblance. Scientia Gerund 23:17–27
Ludeña EV (1987) In: Fraga S (ed) Química Teórica, Nuevas Tendencias, vol 4. CSIC, Madrid, pp 117–160
Löwdin PO (1955) Quantum theory of many-particle systems. I. Physical interpretations by means of density matrices, natural spin-orbitals, and convergence problems in the method of configuration interaction. Phys Rev 97:1474–1489
Löwdin PO (1955) Quantum theory of many-particle systems. II. Study of the ordinary hartree-fock approximation. Phys Rev 97:1490–1508
Löwdin PO (1955) Quantum theory of many-particle systems. III. Extension of the Hartree–Fock scheme to include degenerate systems and correlation effects. Phys Rev 97:1509–1520
March NH (1992) Electron density theory of atoms and molecules. Academic Press, London
Mc Weeny R (1959) Proc Roy Soc A 253:242–259
Mc Weeny R (1960) Some recent advances in density matrix theory. Rev Mod Phys 32:335–369
Mc Weeny R (1978) Methods of molecular quantum mechanics. Academic Press, London
Mestres J, Solà M, Duran M, Carbó R (1994) On the calculation of ab initio quantum molecular similarities for large systems. J Comput Chem 15:1113–1120
Mestres J, Solà M, Duran M, Carbó R (1994) On the use of ab initio quantum molecular similarities as an interpretative tool for the studyof chemical reactions. J Am Chem Soc 116:5909–5915
Mestres J, Solà M, Carbó R (1995) First-order molecular descriptors for molecular steric similarity. Scientia Gerund 21:165–173
Mestres J, Solà M, Carbó R, Luque FJ, Orozco M (1996) Effect of solvation on the charge distribution of a series of anionic, neutral, and cationic species. A quantum molecular similarity study. J Phys Chem 100:606–610
Mezey PG (1993) Shape in chemistry: An introduction to molecular shape and topology. VCH, New York
Mezey PG (1995) Molecular similarity I. In: Sen K (ed) Topics in current chemistry, vol 173. Springer, Berlin, pp 63–83
Mezey PG (1998) Averaged electron densities for averaged conformations. J Comput Chem 19:1337–1344
Mezey PG (2005) Graph representations of molecular similarity measures based on topological resolution. J Comput Methods Sci Eng 5:109–114
Mezey PG, Ponec R, Amat L, Carbó-Dorca R (1999) Quantum similarity approach to the characterization of molecular chirality. Enantiomers 4:371–378
Myers RH (1990) Classical and modern regression with applications. PWS-KENT Publishing company, Boston
Netzeva TI, Gallegos A, Worth AP (2006) Topological quantum similarity indices based on fitted densities: Comparison of the applicability domain of a QSAR for estrogenicity with a large chemical inventory. Environ Toxicol Chem 25:1223–1230
Ortiz JB, Cioslowski J (1991) Chem Phys Lett 185:270–275
Paniagua JC, Alemany P (1999–2000) Química Quàntica. Vols 1 and 2, Barcelona, 1999–2000
Parr RG (1963) The quantum theory of molecular electronic structure. WA Benjamin, New York
Patlewicz G, Jeliazkova N, Gallegos Saliner A, Worth AP (2008) Toxmatch – A new software tool to aid in the development and evaluation of chemically similar groups. SAR QSAR Environ Res 19:397–412
Pauling L, Wilson EB Jr (1985) Introduction to Quantum mechanics. Dover, New York
Petke JD (1993) J Comput Chem 14:928–933
Pierre DA (1969) Optimization theory with applications. Wiley, New York
Pilar FL (1990) Elementary quantum chemistry. McGraw-Hill, Princeton
Pla L (1986) Análisis Multivariado: Método de Componentes Principales Monography no 27. Secretaría General de la OEA, Washington,
Ponec R (1993) J Chem Inf Comput Sci 33:805–811
Ponec R (1995) Overlap determinant method in the theory of pericyclic reactions. Lecture notes in chemistry, vol 65. Springer, Berlin
Ponec R, Strnad M (1990) Collect Czechoslov Chem Commun 55:2583–2589
Ponec R, Strnad M (1990) Collect Czechoslov Chem Commun 55:896–902
Ponec R, Strnad M (1991) J Math Chem 8:103–112
Ponec R, Strnad M (1991) J Phys Organic Chem 4:701–705
Ponec R, Strnad M (1992) Int J Quantum Chem 42:501–508
Ponec R, Amat LL, Carbó-Dorca R (1999) Molecular basis of quantitative structure-properties relationships (QSPR): A quantum similarity approach. J Comput Aided Mol Design 13:259–270
Ponec R, Amat LL, Carbó-Dorca R (1999) Quantum similarity approach to LFER: Substituent and solvent effects on the acidities of carboxylic acids. J Phys Organic Chem 12:447–454
Riera A (1992) J Mol Struct Theochem 259:83–98
Robert D, Carbó-Dorca R (1998) A formal comparison between molecular quantum similarity measures and indices. J Chem Inf Comput Sci 38:469–475
Robert D, Carbó-Dorca R (1998) Analyzing the triple density molecular quantum similarity measures with the INDSCAL model. J Chem Inf Comput Sci 38:620–623
Robert D, Carbó-Dorca R (1998) On the extension of QS to atomic nuclei: Nuclear QS. J Math Chem 23:327–351
Robert D, Carbó-Dorca R (1998) Structure-property relationships in nuclei. Prediction of the binding energy per nucleon usinga quantum similarity approach. Il Nuovo Cimento A111:1311–1321
Robert D, Amat LL, Carbó-Dorca R (1999) Three-dimensional quantitative structure-activity relationships from tuned molecular quantum similarity measures. Prediction of the corticosteroid-binding globulin binding affinity for a steroid family. Chem Inf Comput Sci 39:333–344
Safouhi H (2005) Analytical and numerical development for the two-centre overlap-like quantum similarity integrals over Slater-type functions. J Phys A: Mathematical and General 38:7341–7361
Sen K (1995) Molecular similarity. Topics in current chemistry. vols 173, 174. Springer, Berlin
Shavitt I (1977) In: Schaefer HF III (ed) Methods of electronic structure theory. Modern theoretical chemistry, vol 3. Plenum Press, New York, pp 189–275
Sobolev SL (1938) Math Sb 4:471
Solà M, Mestres J, Duran M, Carbó R (1994) Ab initio quantum molecular similarity measures on metal-substituted carbonic anhydrase (M (II) CA, M=Be, Mg, Mn, Co, Ni, Cu, Zn, and Cd). J Chem Inf Comput Sci 34:1047–1053
Solà M, Mestres J, Carbó R, Duran M (1996) A Comparative analysis by means of Quantum Molecular Similarity Measures of Density Distributions derived from conventional ab initio and Density Functional Methods. J Chem Phys 104:636–647
Solà M, Mestres J, Oliva JM, Duran M, Carbó C (1996) The use of ab initio quantum molecular selfsimilarity measures to analyze electronic charge density distributions. Int J Quantum Chem 58:361–372
Spiegel MR (1968) Mathematical handbook. McGraw-Hill, New York
Stoer J, Witzgall CH (1970) Convexity and optimization in finite dimensions. Die Grundlehren der matematischen Wissenschaften in Einzeldarstellungen, Band 163. Springer, Berlin
Trillas E, Alsina C, Terricabras JM (1995) Introducción a la Lógica Difusa. Ariel Matemática, Barcelona
Tsakovska I, Gallegos A, Netzeva T, Pavan M, Worth AP (2007) Evaluation of SARs for the prediction of eye irritation/corrosion potential-structural inclusion rules in the BfR decision support system. SAR QSAR Environ Res 18:221–235
Vinogradov IM (1989) Encyclopaedia of Mathematics, vol 4. Kluwer Academic, Dordrecht, pp 422–428
Vracko M, Bandelj V, Barbieri P, Benfenati E, Chaudhry Q, Cronin M, Devillers J, Gallegos A, Gini G, Gramatica P, Helma C, Neagu D, Netzeva T, Pavan M, Patlewicz G, Randic M, Tsakovska I, Worth A (2006) Validation of counter propagation neural network models for predictive toxicology according to the OECD principles. A Case Study. SAR QSAR Environ Res 17:265–284
Wagener M, Sadowski J, Gasteiger J (1995) J Am Chem Soc 117:7769–7775
Whitley DC, Ford MG, Livingstone DJ (2000) Unsupervised forward selection: A method for eliminating redundant variables. Chem Inf Comput Sci 40:1160–1168
Wilkinson JH (1965) The algebraic eigenvalue problem. Clarendon Press, Oxford
Wilkinson JH, Reinsch C (1971) Linear álgebra. Springer, Berlin
Worth AP, Bassan A, de Bruijn J, Gallegos A, Netzeva T, Patlewicz G, Pavan M, Tsakovska I, Eisenreich E (2007) The role of the european chemicals bureau in promoting the regulatory use of (Q)SAR methods. SAR QSAR Environ Res 18:111–125
Zadeh LA (1965) Inform Control 8:338
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Carbó-Dorca, R., Gallegos, A. (2009). Quantum Similarity and Quantum Quantitative Structure-Properties Relationships (QQSPR). In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_440
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